6. Trigonometric Identities and More Equations

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.

$\sec^3\theta=\sec\theta+\frac{\tan^2\theta}{\cos\theta}$

Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients appear and all functions are of θ only.

csc θ - sin θ

Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients appear and all functions are of θ only.

csc² θ + sec² θ

Perform each indicated operation and simplify the result so that there are no quotients.

sec x/csc x + csc x/sec x

Work each problem.

Given tan x = -5⁄4, where π/2< x < π, use the trigonometric identities to find cot x, csc x and sec x.

Perform each indicated operation and simplify the result so that there are no quotients.

cos β(sec β + csc β)

Work each problem.

Find the exact values of sin x, cos x, and tan x, for x = π/12 , using

a. difference identities

b. half-angle identities. 

Perform each indicated operation and simplify the result so that there are no quotients.

cos x/sec x + sin x/csc x

For each expression in Column I, use an identity to choose an expression from Column II with the same value. Choices may be used once, more than once, or not at all.

sin 35°

Perform each indicated operation and simplify the result so that there are no quotients.

(tan x + cot x)²

For each expression in Column I, use an identity to choose an expression from Column II with the same value. Choices may be used once, more than once, or not at all.

-sin 35°

For each expression in Column I, choose the expression from Column II that completes an identity.

2. csc x = ____

Perform each indicated operation and simplify the result so that there are no quotients.

(1 + tan θ)² - 2 tan θ

For each expression in Column I, use an identity to choose an expression from Column II with the same value. Choices may be used once, more than once, or not at all.

cos 75°

Perform each indicated operation and simplify the result so that there are no quotients.

1/( sin α - 1) - 1/(sin α + 1)

For each expression in Column I, use an identity to choose an expression from Column II with the same value. Choices may be used once, more than once, or not at all.

sin 300°

Factor each trigonometric expression.

sec² θ - 1

For each expression in Column I, use an identity to choose an expression from Column II with the same value. Choices may be used once, more than once, or not at all.

cos(-55°)

Factor each trigonometric expression.

(tan x + cot x)² - (tan x - cot x)²

Factor each trigonometric expression.

4 tan² β + tan β - 3

Factor each trigonometric expression.

cot⁴ x + 3 cot² x + 2

Factor each trigonometric expression.

sin³ α + cos³ α

Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.

cot α sin α

Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.

cot t tan t

Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.

(csc θ sec θ)/cot θ

For each expression in Column I, choose the expression from Column II that completes an identity. One or both expressions may need to be rewritten.

-tan x cos x

For each expression in Column I, choose the expression from Column II that completes an identity.

4. cot x = ____

Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.

csc² t - 1

For each expression in Column I, choose the expression from Column II that completes an identity. One or both expressions may need to be rewritten.

sec x/csc x

Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.

1/ tan² α + cot α tan α

For each expression in Column I, choose the expression from Column II that completes an identity. One or both expressions may need to be rewritten.

cos² x

Each expression simplifies to a constant, a single function, or a power of a function. Use fundamental identities to simplify each expression.

1 - 1/sec² x

Concept Check Suppose that sec θ = (x+4)/x.

Find an expression in x for tan θ.

Verify that each equation is an identity.

tan α/sec α = sin α

Perform each transformation. See Example 2.

Write cot x in terms of sin x.

Verify that each equation is an identity.

(tan² α + 1)/ sec α = sec α

Perform each transformation. See Example 2.

Write cot x in terms of csc x.

Verify that each equation is an identity.

sin² β (1 + cot² β) = 1

Verify that each equation is an identity.​​

2 cos³ x - cos x = (cos² x - sin² x)/sec x

Perform each transformation. See Example 2.

Write sec x in terms of sin x.

Verify that each equation is an identity.

sin² α + tan² α + cos² α = sec² α

Verify that each equation is an identity.​​

(sin 2x)/(sin x) = 2/sec x

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

tan θ cos θ

Verify that each equation is an identity.

(sin² θ)/cos θ = sec θ - cos θ

Verify that each equation is an identity.​​

(2 tan B)/(sin 2B) = sec² B

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

csc θ cos θ tan θ

Verify that each equation is an identity.

sec⁴ x - sec² x = tan⁴ x + tan² x

Verify that each equation is an identity.​​

(2 cot x)/(tan 2x) = csc² x - 2

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

sin θ sec θ

Verify that each equation is an identity.

(sec α - tan α)² = (1 - sin α)/(1 + sin α)

Verify that each equation is an identity.​​

csc A sin 2A - sec A = cos 2A sec A

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

cot² θ(1 + tan² θ)

For each expression in Column I, choose the expression from Column II that completes an identity.

6. sec² x = ____

Verify that each equation is an identity.

[(sec θ - tan θ)² + 1]/(sec θ csc θ - tan θ csc θ) = 2 tan θ

Verify that each equation is an identity.​​

2 cos² θ - 1 = (1 - tan² θ)/(1 + tan² θ)

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

(sec θ - 1) (sec θ + 1)

Verify that each equation is an identity.

1/(sec α - tan α) = sec α + tan α

Verify that each equation is an identity.​​

sec² α - 1 = (sec 2α - 1)/(sec 2α + 1)

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

1 + cot(-θ)/cot(-θ)

Verify that each equation is an identity.

(csc θ + cot θ)/(tan θ + sin θ) = cot θ csc θ

Verify that each equation is an identity.​​

sin³ θ = sin θ - cos² θ sin θ

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

[1 - sin²(-θ)]/[1 + cot²(-θ)]

Verify that each equation is an identity.

sin² θ (1 + cot² θ) - 1 = 0

Verify that each equation is an identity.​​

2 cos² (x/2) tan x = tan x+ sin x

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

csc θ - sin θ

Verify that each equation is an identity.

(sin⁴ α - cos⁴ α )/(sin² α - cos² α) = 1

Verify that each equation is an identity.​​

(1/2)cot (x/2) - (1/2) tan (x/2) = cot x

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

(sin θ - cos θ) (csc θ + sec θ)

Verify that each equation is an identity.

(cot² t - 1)/(1 + cot² t) = 1 - 2 sin² t

Verify that each equation is an identity.​​

(sin 3t + sin 2t)/(sin 3t - sin 2t ) = tan (5t/2)/(tan (t/2))

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

cos θ (cos θ - sec θ)

Verify that each equation is an identity.

tan² α sin² α = tan² α + cos² α - 1

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

(sec²θ - 1)/(csc²θ - 1)

Verify that each equation is an identity.

sin θ/(1 - cos θ) - sin θ cos θ/( 1 + cos θ) = csc θ (1 + cos² θ)

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

tan(-θ)/sec θ

Verify that each equation is an identity.

(1 + sin θ)/(1 - sin θ) - (1 - sin θ)/( 1 + sin θ) = 4 tan θ sec θ

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

-sec² (-θ) + sin² (-θ) + cos² (-θ)

Verify that each equation is an identity.

sin θ + cos θ = sin θ/(1 - cot θ) + cos θ/(1 - tan θ)

Let csc x = -3. Find all possible values of (sin x + cos x)/sec x.

Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients appear and all functions are of θ only.

cot(-θ)/sec(-θ)

Verify that each equation is an identity.

(1 + sin x + cos x)² = 2(1 + sin x) (1 + cos x)

Verify that each equation is an identity.

(sec α + csc α) (cos α - sin α) = cot α - tan α

Verify that each equation is an identity.

(1 - cos θ)/(1 + cos θ) = 2 csc² θ - 2 csc θ cot θ - 1

Verify that each equation is an identity.

sin² x(1 + cot x) + cos² x(1 - tan x) + cot² x = csc² x

Verify that each equation is an identity.

sin³ θ + cos³ θ = (cos θ + sin θ) (1 - cos θ sin θ)

Find values of the sine and cosine functions for each angle measure.

B, given cos 2B = 1/8 , 540° < 2B < 720°

Find values of the sine and cosine functions for each angle measure.

2y, given sec y = -5/3, sin y > 0

Use the given information to find each of the following.

sin A/2, given cos A/2 = - 3, 90° < A < 180°

Use the given information to find each of the following.

sin y, given cos 2y = -1/3 , π/2 < y < π

Use the given information to find each of the following.

sin 2x, given sin x = 0.6, π/2 < y < π

Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically.

(1 - cos 2x)/sin 2x

Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically.

(cos x sin 2x)/1 + cos 2x)

Graph each expression and use the graph to make a conjecture, predicting what might be an identity. Then verify your conjecture algebraically.

csc x - cot x

Determine whether the positive or negative square root should be selected.

cos 58° = ±√ (1 + cos 116°)/2]

Determine whether the positive or negative square root should be selected.

sin (-10°) = ± √[(1 - cos (-20°))/2]

The half-angle identity

tan A/2 = ± √[(1 - cosA)/(1 + cos A)]

can be used to find tan 22.5° = √(3 - 2√2), and the half-angle identity

tan A/2 = sin A/(1 + cos A)

can be used to find tan 22.5° = √2 - 1. Show that these answers are the same, without using a calculator. (Hint: If a > 0 and b > 0 and a² = b², then a = b.)

Match each expression in Column I with its value in Column II.

8. tan (-π/8)

Match each expression in Column I with its value in Column II.

10. cos 67.5°

Use a half-angle identity to find each exact value.

sin 195°

Use a half-angle identity to find each exact value.

cos 195°

Use a half-angle identity to find each exact value.

sin 165°

Use the given information to find each of the following.

sin x/2 , given cos x = - 5/8, with π/2 < x < π

Use the given information to find each of the following.

cos θ/2 , given sin θ = - 4/5 , with 180° < θ < 270°

Use the given information to find each of the following.

cos x/2 , given cot x = -3, with π/2 < x < π

Use the given information to find each of the following.

cot θ/2, given tan θ = -(√5)/2 , with 90° < θ < 180°

Use the given information to find each of the following.

cos θ, given cos 2θ = 1/2 and θ terminates in quadrant II

Use the given information to find each of the following.

sin x, given cos 2x = 2/3 , with π < x < 3π/2

If cos x = -0.750 and sin ≈ 0.6614, then tan x/2 ≈ .

Simplify each expression.

√[(1 + cos 76°)/2]

Simplify each expression.

√[(1 + cos 165°)/(1 - cos 165°)]

Simplify each expression.

sin 158.2°/(1 + cos 158.2°)

Simplify each expression.

±√[(1 + cos 18x)/2]

Simplify each expression.

±√[(1 + cos 20α)/2]

Simplify each expression.

±√[(1 - cos 8θ)/(1 + cos 8θ)]

Simplify each expression.

±√[(1 - cos 5A)/(1 + cos 5A)]

Simplify each expression.

± √[(1 + cos (x/4))/2]

Simplify each expression.

±√[(1 - cos (3θ/5))/2]

Verify that each equation is an identity.

cot² (x/2) = (1 + cos x)²/(sin² x)

Verify that each equation is an identity.

(sin 2x)/(2sin x) = cos² (x/2) - sin² (x/2)

Verify that each equation is an identity.

tan (θ/2) = csc θ - cot θ

Verify that each equation is an identity.

cos x = (1 - tan² (x/2))/(1 + tan² (x/2))

Advanced methods of trigonometry can be used to find the following exact value.

sin 18° = (√5 - 1)/4

(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.

cos 18°

Advanced methods of trigonometry can be used to find the following exact value.

sin 18° = (√5 - 1)/4

(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.

cot 18°

Advanced methods of trigonometry can be used to find the following exact value.

sin 18° = (√5 - 1)/4

(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.

csc 18°

Advanced methods of trigonometry can be used to find the following exact value.

sin 18° = (√5 - 1)/4

(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.

tan 72°

Advanced methods of trigonometry can be used to find the following exact value.

sin 18° = (√5 - 1)/4

(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.

csc 72°

Advanced methods of trigonometry can be used to find the following exact value.

sin 18° = (√5 - 1)/4

(See Hobson's A Treatise on Plane Trigonometry.) Use this value and identities to find each exact value. Support answers with calculator approximations if desired.

sin 162°