Verify that each equation is an identity.
(csc θ + cot θ...

Question

Verify that each equation is an identity.

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

Accepted Answer

Hello. Today we are going to be verifying whether the given equation is an identity or not. In order to show that the equation is an identity, we need to show that the left hand side of the equation will equal to the right hand side of the equation. Let's go ahead and take a look at the left hand side of the equation. We are given seeking of data plus tangent of the is divided by co tangent of theater plus cosine of the. Now, in order to simplify the left hand side, let's rewrite all of the terms to be in forms of sine and cosine. In order to do this, we're going to be using the trigonometric identities for second cotangent and tangent. The trigonometric identity for seeking of data is given to us as one divided by cosine of the, the trigonometric identity for tangent of data is given to us as tangent of the is equal to sine theta divided by cosine of the and the trigonometric identity for cotangent of data is the inverse of tangent of data which is given to us as cosine of the divided by sine of the using these three trigonometric identities will allow us to rewrite the left hand side of the equation as one divided by cosine of theta plus sine, theta divided by cosine of theta is divided by cosine of data divided by s of theta plus cosine of data. There's a few things that we can do to simplify the left hand side. First, let's go ahead and take a look at the denominator. We need to add the fractions cosine of the divided by sine of the plus cosine of the. In order to add these two quantities together, we need to have the same denominator of sine theta. Currently, we can rewrite the term cosine of theta to be divided by one. And if we were to multiply this fraction by sine theta divided by sine theta, we can rewrite the right hand side or the overall denominator to be cosine of data divided by sine theta plus cosine of the multiplied by sine theta, divided by sine of data. And finally, we can add these two fractions together to give us cosine of data plus cosine of data multiplied by sine of the is all divided by sine of the. So we can go ahead and simplify the denominator one more time by factoring out a cosine in its numerator that will allow us to rewrite the overall denominator to be cosine of the multiply by the quantity one plus sine of the divided by s of the. This is going to be the simplest form of the current denominator. Now, let's take a look at the numerator. Both of the fractions in the numerator have the same denominator of cosine. So we can go ahead and add both of the terms of the numerator to be one plus sine. Theta is divided by cosine of theta. Currently, we have a complex fraction in order to simplify the complex fraction will multiply the denominator by its reciprocal and multiply the numerator by the same quantity, the reciprocal of the denominator will be sine theta divided by cosine of the multiplied by one plus sine of the. And again, we'll be multiplying the same quantity to the overall numerator. This will allow us to reduce the overall denominator to be just one leaving us with one plus sine theta divided by cosine theta multiplied by the quantity sine theta divided by cosine theta multiplied by one plus sine of the. Notice that in the numerator, we have one plus sign of data. And in the denominator, we have the same quantity, we can go ahead and reduce both of these quantities to be just one. And that will allow us to simplify the left hand side to be sine of the divided by cosine of the multiplied by cosine of data. If we were to multiply both cosine data together, we can simplify the denominator to be cosine squared of the. So currently, we have reduced the left hand side of the equation to be just one term. However, notice that in the original equation, the right hand side is given to us as a product of two terms. What this means is that we need to figure out a way to rewrite the left hand side to be two terms. Well, we can represent sine theta divided by cosine squared of theta as a product of two terms, we can rewrite the left hand side to be one divided by cosine of the multiplied by sine of data divided by cosine of data. No, we can represent the left hand side as this product. Because if we were to multiply these two quantities together one multiplied by sine theta will give us sine theta and cosine data multiplied by cosine data will allow us to get cosine squared of data. So both of these quantities will be equivalent to each other. Finally notice that we have the quantities one divided by cosine theta and sine theta divided by cosine theta. The trigonometric identity for sin is one divided by cosine data. And the trigonometric identity for tangent is sine data divided by cosine of the using the trigger to metric identities, we can rewrite the left hand side one more time to be seeking of the multiplied by tangent of data. And this is equivalent to the right hand side of the original equation proving that the given equation is an identity. And what this means is that the answer to this problem is going to be a. So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

Verify that each equation is an identity.
(csc θ + cot θ)/(tan θ + sin θ) = cot θ csc θ

Key Concepts

Trigonometric Identities

Reciprocal Functions

Simplifying Trigonometric Expressions

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