Use the even-odd identities to evaluate the expression. cos ⁡ ( − θ ) − cos ⁡ θ \cos\left(-\theta\right)-\cos\theta cos ( − θ ) − cos θ

− cos ⁡ θ -\cos\theta − cos θ

2 cos ⁡ θ 2\cos\theta 2 cos θ

− 2 cos ⁡ θ -2\cos\theta − 2 cos θ

Use the even-odd identities to evaluate the expression. − cot ⁡ ( θ ) ⋅ sin ⁡ ( − θ ) -\cot\left(\theta\right)\cdot\sin\left(-\theta\right) − cot ( θ ) ⋅ sin ( − θ )

− cos ⁡ θ -\cos\theta − cos θ

cos ⁡ θ sin ⁡ 2 θ \frac{\cos\theta}{\sin^2\theta} s i n 2 θ c o s θ

Select the expression with the same value as the given expression. sec ⁡ ( − 4 π 5 ) \sec\left(-\frac{4\pi}{5}\right) sec ( − 5 4 π )

sec ⁡ ( 4 π 5 ) \sec\left(\frac{4\pi}{5}\right) sec ( 5 4 π )

cos ⁡ ( 4 π 5 ) \cos\left(\frac{4\pi}{5}\right) cos ( 5 4 π )

− cos ⁡ ( 4 π 5 ) -\cos\left(\frac{4\pi}{5}\right) − cos ( 5 4 π )

− sec ⁡ ( 4 π 5 ) -\sec\left(\frac{4\pi}{5}\right) − sec ( 5 4 π )

Select the expression with the same value as the given expression. sin ⁡ ( − 38 ° ) \sin\left(-38\degree\right) sin ( − 38° )

− sin ⁡ 38 ° -\sin38\degree − sin 38°

sin ⁡ 38 ° \sin38\degree sin 38°

− sin ⁡ ( − 38 ° ) -\sin\left(-38\degree\right) − sin ( − 38° )

1 − sin ⁡ 38 ° \frac{1}{-\sin38\degree} − s i n 38° 1

Use the Pythagorean identities to rewrite the expression as a single term. ( 1 + csc ⁡ θ ) ( 1 − csc ⁡ θ ) \left(1+\csc\theta\right)\left(1-\csc\theta\right) ( 1 + csc θ ) ( 1 − csc θ )

− cot ⁡ 2 θ -\cot^2\theta − cot 2 θ

− csc ⁡ 2 θ -\csc^2\theta − csc 2 θ

cot ⁡ 2 θ \cot^2\theta cot 2 θ

Use the Pythagorean identities to rewrite the expression with no fraction. 1 1 − sec ⁡ θ \frac{1}{1-\sec\theta} 1 − s e c θ 1

− cot ⁡ 2 θ ( 1 + sec ⁡ θ ) -\cot^2\theta\left(1+\sec\theta\right) − cot 2 θ ( 1 + sec θ )

1 + sec ⁡ θ 1+\sec\theta 1 + sec θ

1 tan ⁡ 2 θ \frac{1}{\tan^2\theta} t a n 2 θ 1

− tan ⁡ 2 θ ( 1 + sec ⁡ θ ) -\tan^2\theta\left(1+\sec\theta\right) − tan 2 θ ( 1 + sec θ )

Simplify the expression. tan ⁡ 2 θ − sec ⁡ 2 θ + 1 \tan^2\theta-\sec^2\theta+1 tan 2 θ − sec 2 θ + 1

csc ⁡ 2 θ + 1 \csc^2\theta+1 csc 2 θ + 1

Simplify the expression. tan ⁡ ( − θ ) sec ⁡ ( − θ ) \frac{\tan\left(-\theta\right)}{\sec\left(-\theta\right)} s e c ( − θ ) t a n ( − θ )

− sin ⁡ θ -\sin\theta − sin θ

− cot ⁡ θ -\cot\theta − cot θ

Identify the most helpful first step in verifying the identity. sec ⁡ 3 θ = sec ⁡ θ + tan ⁡ 2 θ cos ⁡ θ \sec^3\theta=\sec\theta+\frac{\tan^2\theta}{\cos\theta} sec 3 θ = sec θ + c o s θ t a n 2 θ

Use reciprocal identity to rewrite sec ⁡ θ \sec\theta sec θ on right side of equation.

Subtract sec ⁡ θ \sec\theta sec θ from both sides.

Rewrite tan ⁡ 2 θ \tan^2\theta tan 2 θ in terms of sine and cosine.

6. Trigonometric Identities and More Equations

Introduction to Trigonometric Identities

6. Trigonometric Identities and More Equations

Introduction to Trigonometric Identities: Videos & Practice Problems

Video Lessons Practice Worksheet

Topic summary

Understanding the classification of trigonometric functions as even or odd is crucial for simplifying expressions. The cosine function is even, meaning $f (- x) = f (x)$ , while the sine function is odd, leading to $f (- x) = - f (x)$ . Pythagorean identities, such as $\sin^{2} (θ) + \cos^{2} (θ) = 1$ , facilitate the simplification of trigonometric expressions, ensuring a solid grasp of these concepts is essential for solving complex problems effectively.

concept

Even and Odd Identities

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Even and Odd Identities Video Summary

Understanding the classification of trigonometric functions as even or odd is essential for simplifying expressions and solving problems effectively. An even function, such as cosine, satisfies the property that f(-x) = f(x). This means that the graph of an even function is symmetric about the y-axis. For example, the cosine function evaluated at $\frac{\pi}{2}$ yields 0, and similarly, f(-x) = -f(x). This indicates that the graph of an odd function is symmetric about the origin. For instance, evaluating sine at $\frac{\pi}{2}$ gives 1, while cos(-$\theta$) = cos($\theta$), while the sine and tangent of a negative angle can be expressed as sin(-$\theta$) = -sin($\theta$) and tan(-$\theta$) = -tan($\theta$), respectively. These identities simplify calculations significantly.

For instance, to evaluate cos(-$\frac{\pi}{4}$), we can use the even identity to find that it equals cos($\frac{\pi}{4}$), which is csc(-$\frac{\pi}{6}$), we can rewrite it as 1/sin(-$\frac{\pi}{6}$), leading to

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example

Example 1

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Example 1 Video Summary

Understanding even and odd identities is crucial for simplifying trigonometric expressions, especially when dealing with negative arguments. These identities allow us to rewrite expressions in a more manageable form, often eliminating negative angles entirely.

For instance, consider the expression involving the negative tangent of negative theta, expressed as -tan(-$\theta$). According to the even-odd identity for tangent, we know that tan(-$\theta$) = -tan($\theta$). By substituting this identity into our expression, we have:

-tan(-$\theta$) = -(-tan($\theta$)) = tan($\theta$).

This simplification shows that two negatives result in a positive, allowing us to express the original function without any negative arguments.

Next, let’s examine the expression sin(-$\theta$) / cos(-$\theta$). Here, we apply the even-odd identities for sine and cosine. We know that:

sin(-$\theta$) = -sin($\theta$)
cos(-$\theta$) = cos($\theta$)

Substituting these identities into our expression gives:

sin(-$\theta$) / cos(-$\theta$) = -sin($\theta$) / cos($\theta$).

Recognizing that sin($\theta$) / cos($\theta$) is equivalent to tan($\theta$), we can further simplify this to:

-tan($\theta$).

Both methods of simplification lead to the same result, demonstrating that there are often multiple approaches to arrive at the correct answer. Mastering these identities not only aids in simplifying expressions but also enhances overall understanding of trigonometric functions.

Problem

Use the even-odd identities to evaluate the expression.
$\cos\]\left\)(-$\theta\[\right$)-\(\cos\]\theta$

$-$\cos\]\theta$$

$2$\cos\]\theta$$

$-2$\cos\]\theta$$

Problem

Use the even-odd identities to evaluate the expression.
$-$\cot\]\left$($\theta\[\right$)$\cdot\]\sin\[\left$(-$\theta\]\right$)$

$\tan\]\theta$

$-$\cos\]\theta$$

$\cos\]\theta$

$\frac{\cos\theta}{\sin^2\theta}$

Problem

Select the expression with the same value as the given expression.
$$\sec\[\left$(-$\frac{4\pi}{5}\]\right$)$

$$\cos\[\left$($\frac{4\pi}{5}\]\right$)$

$-$\cos\[\left$($\frac{4\pi}{5}\]\right$)$

$$\sec\[\left$($\frac{4\pi}{5}\]\right$)$

$-$\sec\[\left$($\frac{4\pi}{5}\]\right$)$

Problem

Select the expression with the same value as the given expression.
$$\sin\[\left$(-38$\degree\]\right$)$

$\sin\)38\(\degree$

$-$\sin$38$\degree$$

$-$\sin\[\left$(-38$\degree\]\right$)$

$\frac{1}{-\sin38\degree}$

concept

Pythagorean Identities

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Pythagorean Identities Video Summary

Understanding trigonometric expressions can be simplified significantly by utilizing Pythagorean identities. A fundamental identity states that the sine squared of an angle plus the cosine squared of that angle equals one. This can be expressed mathematically as:

\[\sin^2(\theta) + \cos^2(\theta) = 1\]

This identity is derived from the Pythagorean theorem, which relates the sides of a right triangle. In the context of the unit circle, if we consider a triangle with a hypotenuse of 1, the lengths of the sides can be represented as the sine and cosine of an angle. For example, for the angle $ \frac{\pi}{6} $, we find:

\[\sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \quad \text{and} \quad \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}\]

Squaring these values gives:

\[\left(\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{1}{4} + \frac{3}{4} = 1\]

This confirms that the identity holds true for this angle and, by extension, for any angle.

In addition to the primary identity, two other Pythagorean identities can be derived by manipulating the first. Dividing the first identity by cosine yields:

\[\tan^2(\theta) + 1 = \sec^2(\theta)\]

Similarly, dividing by sine gives:

\[1 + \cot^2(\theta) = \csc^2(\theta)\]

These identities are useful for simplifying expressions involving squared trigonometric functions. For instance, if you encounter the expression:

\[\sec^2(\theta) - \tan^2(\theta)\]

Using the second Pythagorean identity, we can rewrite it as:

\[\sec^2(\theta) - \tan^2(\theta) = 1\]

Another example involves the expression:

\[1 - \cos(\theta) \cdot 1 + \cos(\theta)\]

By applying the difference of squares, this simplifies to:

\[1 - \cos^2(\theta)\]

Using the first Pythagorean identity, we can express this as:

\[1 - \cos^2(\theta) = \sin^2(\theta)\]

Thus, the expression simplifies to:

\[1 - \cos^2(\theta) = \sin^2(\theta)\]

By practicing these identities and their applications, you will become proficient in simplifying trigonometric expressions, making complex problems more manageable.

example

Example 2

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Example 2 Video Summary

In this problem, we explore how to rewrite the expression $\frac{1}{1 + \cos(\theta)}$ using Pythagorean identities to eliminate the fraction. The first step involves multiplying the denominator by $1 - \cos(\theta)$, which is a strategic move that utilizes the difference of squares. This results in:

$1 + \cos(\theta)$ $\times$ $1 - \cos(\theta) = 1 - \cos^2(\theta)$.

Recognizing that $1 - \cos^2(\theta)$ can be rewritten using the first Pythagorean identity, we find that:

$\sin^2(\theta) = 1 - \cos^2(\theta)$.

Substituting this back into our expression, we replace the denominator with $\sin^2(\theta)$, leading to:

$\frac{1 - \cos(\theta)}{\sin^2(\theta)}$.

To eliminate the fraction, we can express this as:

$\frac{1}{\sin^2(\theta)} \times (1 - \cos(\theta))$.

Since $\frac{1}{\sin(\theta)}$ is defined as $\csc(\theta)$, we find that:

$\frac{1}{\sin^2(\theta)} = \csc^2(\theta)$.

Thus, the final expression becomes:

$\csc^2(\theta) \times (1 - \cos(\theta))$.

This transformation successfully eliminates the fraction, providing a clear and simplified expression. The key takeaway is the effective use of Pythagorean identities and algebraic manipulation to achieve the desired form.

Problem

Use the Pythagorean identities to rewrite the expression as a single term.
$$\left$(1+$\csc\[\theta\]\right$)$\left$(1-$\csc\[\theta\]\right$)$

$-$\csc$^2$\theta$$

$\cot\)^2\(\theta$

$-$\cot$^2$\theta$$

Problem

Use the Pythagorean identities to rewrite the expression with no fraction.
$\frac{1}{1-\sec\theta}$

$1+$\sec\]\theta$$

$\frac{1}{\tan^2\theta}$

$-$\cot$^2$\theta\]\left$(1+$\sec\[\theta\]\right$)$

$-$\tan$^2$\theta\]\left$(1+$\sec\[\theta\]\right$)$

concept

Simplifying Trig Expressions

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Simplifying Trig Expressions Video Summary

Understanding how to simplify trigonometric expressions is essential in trigonometry. A fully simplified trigonometric expression must meet three criteria: all arguments should be positive, the expression should contain no fractions, and it should have as few trigonometric functions as possible.

To illustrate this, consider the expression tan(-$\theta$) $\cdot$ csc($\theta$). The first step is to ensure all arguments are positive. Using the even-odd identities, we know that tan(-$\theta$) = -tan($\theta$). Thus, we can rewrite the expression as -tan($\theta$) $\cdot$ csc($\theta$).

Next, we need to reduce the number of trigonometric functions. The tangent and cosecant can be expressed in terms of sine and cosine: tan($\theta$) = $\frac{\sin(\theta)}{\cos(\theta)}$ and csc($\theta$) = $\frac{1}{\sin(\theta)}$. Substituting these identities gives us:

- $\frac{\sin(\theta)}{\cos(\theta)}$ $\cdot$ $\frac{1}{\sin(\theta)}$ = -$\frac{1}{\cos(\theta)}$

Recognizing that -$\frac{1}{\cos(\theta)}$ = -sec($\theta$), we find that the expression simplifies to -sec($\theta$). This meets all criteria for simplification.

In another example, consider the expression $\frac{\sin^2(\theta)}{1 + \cos(\theta)}$. Here, we start with a fraction, so we need to eliminate it. To do this, we can multiply both the numerator and denominator by 1 - $\cos$($\theta$), which is the conjugate of the denominator:

$\frac{\sin^2(\theta)(1 - \cos(\theta))}{(1 + \cos(\theta))(1 - \cos(\theta))}$

The denominator simplifies to 1 - $\cos$^2($\theta$), which can be replaced using the Pythagorean identity $\sin$^2($\theta$) + $\cos$^2($\theta$) = 1. Thus, 1 - $\cos$^2($\theta$) = $\sin$^2($\theta$). This gives us:

$\frac{\sin^2(\theta)(1 - \cos(\theta))}{\sin^2(\theta)}$

After canceling $\sin$^2($\theta$) from the numerator and denominator, we are left with 1 - $\cos$($\theta$). This expression has positive arguments, contains no fractions, and has only one trigonometric function, thus fully simplified.

In summary, simplifying trigonometric expressions involves recognizing identities, eliminating fractions, and reducing the number of functions. There are often multiple valid approaches to achieve simplification, so exploring different methods can enhance understanding and flexibility in problem-solving.

example

Example 3

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Example 3 Video Summary

To simplify the expression $\frac{\sin^2(\theta) - \tan^2(\theta)}{\sin(\theta) + \tan(\theta)}$, we start by recognizing that the numerator can be factored as a difference of squares. The difference of squares identity states that $a^2 - b^2 = (a + b)(a - b)$. Here, we can identify $a = \sin(\theta)$ and $b = \tan(\theta)$, leading to:

$\sin^2(\theta) - \tan^2(\theta) = (\sin(\theta) + \tan(\theta))(\sin(\theta) - \tan(\theta))$.

Substituting this back into our expression gives:

$\frac{(\sin(\theta) + \tan(\theta))(\sin(\theta) - \tan(\theta))}{\sin(\theta) + \tan(\theta)}$.

Since the term $(\sin(\theta) + \tan(\theta))$ appears in both the numerator and the denominator, we can cancel it out, provided that $\sin(\theta) + \tan(\theta) \neq 0$. This simplification results in:

$\sin(\theta) - \tan(\theta)$.

At this point, we have eliminated the fraction and reduced the expression to a simpler form. The final simplified expression is:

$\sin(\theta) - \tan(\theta)$.

This expression contains two trigonometric functions, which is the minimum number we can achieve given the nature of the operations involved. Thus, we have successfully simplified the original expression.

example

Example 4

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Example 4 Video Summary

To simplify the expression involving trigonometric functions, we start with the expression:

\[\frac{\cos(\theta) + \csc(\theta)}{\cos(\theta) + \sin(\theta) - \frac{\sec(\theta)}{\sin(\theta)}}\]

Our goal is to simplify this expression by eliminating fractions and reducing the number of trigonometric functions. First, we identify the common denominator for the two fractions in the numerator and denominator. The common denominator is the product of the sine and cosine functions, or \[\sin(\theta) \cos(\theta)\].

To combine the fractions, we multiply the first fraction by \[\frac{\sin(\theta)}{\sin(\theta)}\] and the second fraction by \[\frac{\cos(\theta)}{\cos(\theta)}\]. This gives us:

\[\frac{\sin(\theta) \cos(\theta) + \sin(\theta) \cdot \csc(\theta)}{\sin(\theta) \cos(\theta) + \sin(\theta) \cdot \cos(\theta) - \cos(\theta) \cdot \sec(\theta)}\]

Next, we apply the identities \[\csc(\theta) = \frac{1}{\sin(\theta)}\] and \[\sec(\theta) = \frac{1}{\cos(\theta)}\] to rewrite the expression:

\[\frac{\sin(\theta) \cos(\theta) + \sin(\theta) \cdot \frac{1}{\sin(\theta)}}{\sin(\theta) \cos(\theta) + \sin(\theta) \cos(\theta) - \cos(\theta) \cdot \frac{1}{\cos(\theta)}}\]

After substituting, we simplify the numerator and denominator:

Numerator: \[\sin(\theta) \cos(\theta) + 1\]

Denominator: \[\sin(\theta) \cos(\theta) + \sin(\theta) \cos(\theta) - 1\]

Now, we can cancel terms. The sine in the numerator cancels with the sine in the denominator, and the cosine in the numerator cancels with the cosine in the denominator:

\[\frac{1}{\sin(\theta) \cos(\theta)}\]

Next, we notice that the numerator simplifies to \[2 \sin(\theta) \cos(\theta)\], which can be expressed using the double angle identity:

\[\sin(2\theta) = 2 \sin(\theta) \cos(\theta)\]

Thus, the expression simplifies to:

\[\frac{2 \sin(\theta) \cos(\theta)}{\sin(\theta) \cos(\theta)} = 2\]

In conclusion, the original expression simplifies to a final result of:

\[2\]

This process illustrates the importance of recognizing trigonometric identities and common denominators in simplifying complex expressions.

Problem

Simplify the expression.
$$\tan$^2$\theta$-$\sec$^2$\theta$+1$

$$\csc$^2$\theta$+1$

Problem

Simplify the expression.
$\frac{\tan\left(-\theta\right)}{\sec\left(-\theta\right)}$

$\sin\]\theta$

$-$\sin\]\theta$$

$-$\cot\]\theta$$

Problem

Simplify the expression.
$$\left$($\frac{\tan^2\theta}{\sin^2\theta}$-1$\right$)$\csc$^2$\left$($\theta\[\right$)$\cos$^2$\left$(-$\theta\]\right$)$

$\cot\)^2\(\theta$

$\tan\]\theta$

– 1

concept

Verifying Trig Equations as Identities

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Verifying Trig Equations as Identities Video Summary

Verifying trigonometric identities involves demonstrating that two sides of an equation are equal by simplifying one or both sides. This process relies heavily on recognizing and applying trigonometric identities, particularly Pythagorean identities, which are foundational in trigonometry.

For instance, consider the identity:

\[\frac{\sin \theta \cdot \cos \theta}{1 - \cos^2 \theta} = \frac{1}{\tan \theta}\]

To verify this, start with the more complex side, which is the left side in this case. The denominator, $1 - \cos^2 \theta$, can be simplified using the Pythagorean identity $ \sin^2 \theta + \cos^2 \theta = 1$. Rearranging gives $1 - \cos^2 \theta = \sin^2 \theta$. Thus, the left side simplifies to:

\[\frac{\sin \theta \cdot \cos \theta}{\sin^2 \theta} = \frac{\cos \theta}{\sin \theta} = \cot \theta\]

Recognizing that $\cot \theta = \frac{1}{\tan \theta}$, we conclude that both sides are equal, verifying the identity.

In another example, consider the identity:

\[\frac{\sec^2 \theta - \tan^2 \theta}{\cos(-\theta) + 1} = \frac{1 - \cos \theta}{\sin^2 \theta}\]

Again, start with the left side. The numerator $\sec^2 \theta - \tan^2 \theta$ can be simplified using the identity $\sec^2 \theta = \tan^2 \theta + 1$, leading to a simplification of the numerator to 1. The denominator simplifies as $\cos(-\theta) = \cos \theta$, resulting in:

\[\frac{1}{\cos \theta + 1}\]

Now, simplifying the right side involves multiplying the numerator and denominator by $1 + \cos \theta$ to utilize the difference of squares, yielding:

\[\frac{1 - \cos^2 \theta}{\sin^2 \theta (1 + \cos \theta)} = \frac{\sin^2 \theta}{\sin^2 \theta (1 + \cos \theta)} = \frac{1}{1 + \cos \theta}\]

Both sides are now equal, confirming the identity.

When verifying identities, it is crucial to remain vigilant for opportunities to apply known identities and to simplify expressions methodically. If you encounter difficulties, revisiting the problem from a fresh perspective can often reveal new insights. With practice, your proficiency in verifying trigonometric identities will improve significantly.

example

Example 5

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Example 5 Video Summary

To verify the identity given by the equation:

\[\frac{1 - \sin(\theta)}{\cos(\theta)} - \frac{\cos(\theta)}{1 + \sin(\theta)} = 0\]

we will focus on simplifying the left side. The first step involves finding a common denominator for the two fractions. The denominators are $\cos(\theta)$ and $1 + \sin(\theta)$, so the common denominator will be $\cos(\theta)(1 + \sin(\theta))$.

Next, we rewrite each fraction with this common denominator:

\[\frac{(1 - \sin(\theta))(1 + \sin(\theta))}{\cos(\theta)(1 + \sin(\theta))} - \frac{\cos^2(\theta)}{\cos(\theta)(1 + \sin(\theta)} \]

Multiplying the first fraction's numerator by $1 + \sin(\theta)$ gives:

\[1 - \sin^2(\theta)\]

For the second fraction, multiplying the numerator by $\cos(\theta)$ results in:

\[-\cos^2(\theta)\]

Now, we combine the numerators:

\[1 - \sin^2(\theta) - \cos^2(\theta)\]

Using the Pythagorean identity, which states that $ \sin^2(\theta) + \cos^2(\theta) = 1$, we can substitute $1 - \sin^2(\theta)$ with $\cos^2(\theta)$. Thus, we have:

\[\cos^2(\theta) - \cos^2(\theta) = 0\]

Now, placing this over the common denominator gives:

\[\frac{0}{\cos(\theta)(1 + \sin(\theta))} = 0\]

Since both sides of the original equation equal zero, we have successfully verified the identity:

\[0 = 0\]

This confirms that the identity holds true. If you have any questions, feel free to ask!

example

Example 6

Video duration:

Example 6 Video Summary

To verify the trigonometric identity given by the equation:

\[\sec(\theta) \cdot (1 - \sin^2(\theta)) = \frac{1 + \tan^2(\theta) \cdot \cot^2(\theta)}{\csc^2(\theta) \cdot \sec(\theta)}\]

we start with the more complex right side of the equation. The first step involves recognizing that $1 + \tan^2(\theta)$ can be simplified using the Pythagorean identity:

\[1 + \tan^2(\theta) = \sec^2(\theta)\]

Substituting this into the equation gives us:

\[\frac{\sec^2(\theta) \cdot \cot^2(\theta)}{\csc^2(\theta) \cdot \sec(\theta)}\]

Next, we can cancel one $\sec(\theta)$ from the numerator and denominator, resulting in:

\[\frac{\sec(\theta) \cdot \cot^2(\theta)}{\csc^2(\theta)}\]

Now, we express everything in terms of sine and cosine. Recall that:

$\sec(\theta) = \frac{1}{\cos(\theta)}$
$\cot^2(\theta) = \frac{\cos^2(\theta)}{\sin^2(\theta)}$
$\csc^2(\theta) = \frac{1}{\sin^2(\theta)}$

Substituting these definitions into our expression yields:

\[\frac{\frac{1}{\cos(\theta)} \cdot \frac{\cos^2(\theta)}{\sin^2(\theta)}}{\frac{1}{\sin^2(\theta)}}\]

When we simplify this, we multiply by the reciprocal of the denominator:

\[\frac{1}{\cos(\theta)} \cdot \frac{\cos^2(\theta)}{\sin^2(\theta)} \cdot \sin^2(\theta) = \frac{\cos(\theta)}{1} = \cos(\theta)\]

Now, we turn our attention to the left side of the original identity:

\[\sec(\theta) \cdot (1 - \sin^2(\theta))\]

Using the Pythagorean identity again, we know that:

\[1 - \sin^2(\theta) = \cos^2(\theta)\]

Thus, we can rewrite the left side as:

\[\sec(\theta) \cdot \cos^2(\theta)\]

Substituting $\sec(\theta)$ gives us:

\[\frac{1}{\cos(\theta)} \cdot \cos^2(\theta) = \cos(\theta)\]

Both sides of the equation simplify to $\cos(\theta)$, confirming that the identity is verified:

\[\cos(\theta) = \cos(\theta)\]

Problem

Identify the most helpful first step in verifying the identity.
$$\left$($\frac{\tan^2\theta}{\sin^2\theta}$-1$\right$)=$\sec$^2$\theta\[\sin$^2$\left$(-$\theta\]\right$)$

Add the terms on the left side using a common denominator.

Rewrite left side of equation in terms of sine and cosine.

Use even-odd identity to eliminate negative argument on right side of equation.

Rewrite right side of equation in terms of sine and cosine.

Problem

Identify the most helpful first step in verifying the identity.
$\sec\)^3$\theta$=$\sec\]\theta$+\(\frac{\tan^2\theta}{\cos\theta}$

Rewrite left side of equation in terms of sine and cosine.

Subtract $\sec\]\theta$ from both sides.

Use reciprocal identity to rewrite $\sec\]\theta$ on right side of equation.

Rewrite $\tan\)^2\(\theta$ in terms of sine and cosine.

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Even and odd trigonometric identities help simplify expressions involving negative angles. An even function is symmetric about the y-axis, meaning $f (- x) = f (x)$ . Cosine and secant are even functions. An odd function is symmetric about the origin, meaning $f (- x) = - f (x)$ . Sine, tangent, cosecant, and cotangent are odd functions. These identities are useful for simplifying expressions by replacing trigonometric functions of negative angles with their positive counterparts or negatives, depending on whether the function is even or odd.

Pythagorean identities are derived from the Pythagorean theorem and are used to simplify expressions involving squared trigonometric functions. The primary identity is $\sin^{2} θ + \cos^{2} θ = 1$ . By dividing this identity by either $\cos$ or $\sin$ , you can derive two additional identities: $\tan^{2} θ + 1 = \sec^{2} θ$ and $1 + \cot^{2} θ = \csc^{2} θ$ . These identities allow you to simplify expressions by recognizing and substituting equivalent forms, reducing complexity and aiding in solving trigonometric equations.

Simplifying trigonometric expressions involves several strategies. First, ensure all arguments are positive by using even-odd identities. Second, reduce the number of trigonometric functions by converting them to sine and cosine, as these are the fundamental functions. Third, eliminate fractions by manipulating expressions, such as multiplying by conjugates. Constantly scan for identities, like Pythagorean identities, to substitute and simplify expressions. These strategies help achieve a fully simplified expression, which should have positive arguments, no fractions, and as few trigonometric functions as possible.

Verifying trigonometric identities involves simplifying one or both sides of an equation to demonstrate equality. Start by simplifying the more complicated side using known identities, such as Pythagorean and even-odd identities. Convert functions to sine and cosine to facilitate simplification. The goal is to show that both sides of the equation are equal. This process often requires manipulating expressions, recognizing equivalent forms, and using algebraic techniques. Practice and familiarity with identities are key to successfully verifying trigonometric identities.

Trigonometric identities are crucial in solving problems as they provide relationships between trigonometric functions that simplify complex expressions and equations. They allow for the transformation of expressions, making it easier to solve equations, evaluate functions, and verify identities. Understanding these identities aids in recognizing patterns and equivalent forms, which is essential in calculus, physics, and engineering applications. Mastery of trigonometric identities enhances problem-solving skills and mathematical reasoning, making them indispensable tools in various fields.

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Introduction to Trigonometric Identities: Videos & Practice Problems

Even and Odd Identities

Even and Odd Identities Video Summary

Example 1

Example 1 Video Summary

Use the even-odd identities to evaluate the expression.cos(−θ)−cosθ\(\cos\]\left\)(-\(\theta\[\right\))-\(\cos\]\theta\)cos(−θ)−cosθ

Use the even-odd identities to evaluate the expression.−cot(θ)⋅sin(−θ)-\(\cot\]\left\)(\(\theta\[\right\))\(\cdot\]\sin\[\left\)(-\(\theta\]\right\))−cot(θ)⋅sin(−θ)

Select the expression with the same value as the given expression.sec(−4π5)\(\sec\[\left\)(-\(\frac{4\pi}{5}\]\right\))sec(−54π)

Select the expression with the same value as the given expression.sin(−38°)\(\sin\[\left\)(-38\(\degree\]\right\))sin(−38°)

Pythagorean Identities

Pythagorean Identities Video Summary

Example 2

Example 2 Video Summary

Use the Pythagorean identities to rewrite the expression as a single term.(1+cscθ)(1−cscθ)\(\left\)(1+\(\csc\[\theta\]\right\))\(\left\)(1-\(\csc\[\theta\]\right\))(1+cscθ)(1−cscθ)

Use the Pythagorean identities to rewrite the expression with no fraction.11−secθ\(\frac{1}{1-\sec\theta}\)1−secθ1

Simplifying Trig Expressions

Simplifying Trig Expressions Video Summary

Example 3

Example 3 Video Summary

Example 4

Example 4 Video Summary

Simplify the expression.tan2θ−sec2θ+1\(\tan\)^2\(\theta\)-\(\sec\)^2\(\theta\)+1tan2θ−sec2θ+1

Simplify the expression.tan(−θ)sec(−θ)\(\frac{\tan\left(-\theta\right)}{\sec\left(-\theta\right)}\)sec(−θ)tan(−θ)

Simplify the expression.(tan2θsin2θ−1)csc2(θ)cos2(−θ)\(\left\)(\(\frac{\tan^2\theta}{\sin^2\theta}\)-1\(\right\))\(\csc\)^2\(\left\)(\(\theta\[\right\))\(\cos\)^2\(\left\)(-\(\theta\]\right\))(sin2θtan2θ−1)csc2(θ)cos2(−θ)

Verifying Trig Equations as Identities

Verifying Trig Equations as Identities Video Summary

Example 5

Example 5 Video Summary

Example 6

Example 6 Video Summary

Identify the most helpful first step in verifying the identity.(tan2θsin2θ−1)=sec2θsin2(−θ)\(\left\)(\(\frac{\tan^2\theta}{\sin^2\theta}\)-1\(\right\))=\(\sec\)^2\(\theta\[\sin\)^2\(\left\)(-\(\theta\]\right\))(sin2θtan2θ−1)=sec2θsin2(−θ)

Identify the most helpful first step in verifying the identity.sec3θ=secθ+tan2θcosθ\(\sec\)^3\(\theta\)=\(\sec\]\theta\)+\(\frac{\tan^2\theta}{\cos\theta}\)sec3θ=secθ+cosθtan2θ

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Here’s what students ask on this topic:

What are the even and odd trigonometric identities?

What are the even and odd trigonometric identities?

How do you use Pythagorean identities to simplify trigonometric expressions?

How do you use Pythagorean identities to simplify trigonometric expressions?

What strategies can be used to simplify trigonometric expressions?

What strategies can be used to simplify trigonometric expressions?

How do you verify trigonometric identities?

How do you verify trigonometric identities?

What is the importance of trigonometric identities in solving problems?

What is the importance of trigonometric identities in solving problems?

Your Trigonometry tutors

Use the even-odd identities to evaluate the expression.
$\cos\]\left\)(-\(\theta\[\right\))-\(\cos\]\theta$

Use the even-odd identities to evaluate the expression.
$-\(\cot\]\left\)(\(\theta\[\right\))\(\cdot\]\sin\[\left\)(-\(\theta\]\right\))$

Select the expression with the same value as the given expression.
$\(\sec\[\left\)(-\(\frac{4\pi}{5}\]\right\))$

Select the expression with the same value as the given expression.
$\(\sin\[\left\)(-38\(\degree\]\right\))$

Use the Pythagorean identities to rewrite the expression as a single term.
$\(\left\)(1+\(\csc\[\theta\]\right\))\(\left\)(1-\(\csc\[\theta\]\right\))$

Use the Pythagorean identities to rewrite the expression with no fraction.
$\frac{1}{1-\sec\theta}$

Simplify the expression.
$\(\tan\)^2\(\theta\)-\(\sec\)^2\(\theta\)+1$

Simplify the expression.
$\frac{\tan\left(-\theta\right)}{\sec\left(-\theta\right)}$

Simplify the expression.
$\(\left\)(\(\frac{\tan^2\theta}{\sin^2\theta}\)-1\(\right\))\(\csc\)^2\(\left\)(\(\theta\[\right\))\(\cos\)^2\(\left\)(-\(\theta\]\right\))$

Identify the most helpful first step in verifying the identity.
$\(\left\)(\(\frac{\tan^2\theta}{\sin^2\theta}\)-1\(\right\))=\(\sec\)^2\(\theta\[\sin\)^2\(\left\)(-\(\theta\]\right\))$

Identify the most helpful first step in verifying the identity.
$\sec\)^3\(\theta\)=\(\sec\]\theta\)+\(\frac{\tan^2\theta}{\cos\theta}$