Verify that each equation is an identity.
sin² β (1 + co...

Question

Verify that each equation is an identity.

sin² β (1 + cot² β) = 1

Accepted Answer

Hello. Today we're going to be showing whether the given equation is an identity or not. In order to prove that an equation is an identity, we need to show that the left hand side of the equation is equal to the right hand side of the equation. Let's go ahead and take a look at what is given to us. We are told that cose and squared of X multiplied by the quantity one minus cosine squared of X is equal to one. Let's go ahead and see if we can first start by reducing the quantity one minus cosine squared of X. Recall that we have the Pythagorean identity sine squared of X plus cosine squared of X is equal to one. If we were to subtract cosine to both sides of this equation, we can rewrite the identity as sine squared of X is equal to one minus cosine squared of X. Notice that the right hand side of this identity matches the given quantity in the original equation. What this means is that we can rewrite one minus cosine squared X as sine squared X. So the new equation that we are left with is co sequence squared squared of X multiplied by sine squared of X is equal to one. Now, in order to further simplify this recall that there is a trigonometric identity for cos it that relates it to sign the trigonometric identity for cosecant is coin of X is equal to one divided by sine square by I'm sorry, sin of X. Now, currently, we cannot replace anything using this identity because we have COK and squared X within the original equation. And the identity only uses coin of X. However, if we were to square both sides of this identity, we can rewrite the identity as coin squared of X is equal to one divided by sine squared of X. If we were to replace Kosekin squared X with the new value using the identity, we can rewrite the equation as one divided by sine squared of X multiplied by sine squared of X is equal to one. The left hand side will allow us to reduce both of the sine squared X to be just one. And that will allow us to simplify the equation to be one is equal to one, which is a true statement. This verifies that the given equation is an identity. Therefore, 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.
sin² β (1 + cot² β) = 1

Key Concepts

Pythagorean Identity

Cotangent Function

Trigonometric Identities

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