Verify that each equation is an identity.
tan² α sin² α ...

Question

Verify that each equation is an identity.

tan² α sin² α = tan² α + cos² α - 1

Accepted Answer

Hello, everyone. We are asked if the given equation is an identity, we are given the cotangent squared of X multiplied by the cosine squared of X equals the cotangent squared of X plus the sine squared of X minus one. Our answer choices are a yes B no. First, we need to understand that for this to be an identity, we need to make the left hand side match the right hand side. So I'm going to start with the left hand side. And here I have the cotangent squared of X multiplied by the cosine squared of X. So I recall that we have a pythagorean identity that states that the cosine squared of X is equal to one minus the sine squared of X. I'm gonna use that here. So I'll have the cotangent squared of X multiplied by one minus the sine squared of X. I will distribute the cotangent squared of X into the parentheses. So I have cotangent squared X minus cotangent squared X multiplied by the sine squared of X. I'm going to leave the first cotangent squared of X alone. I'm going to make the second one into the cosine squared of X divided by the sine squared of X. And that'll be multiplied by the sine squared of X. Here in my multiplication, I can cross reduce the sine squared of X in the denominator with the sine squared of X in the numerator. This will give me the cotangent squared of X minus the cosine squared of X. And I'm gonna apply that identity. We mentioned earlier the pythagorean identity where the cosine squared of X equals one minus the sine squared of X again. So we'll have the cotangent squared of X minus in parentheses, one minus the sine squared of X, distributing my negative sign into the parentheses. I get the cotangent squared of X minus one plus the sine squared of X. And if I rearrange this, I will get the cotangent squared of X plus the sine squared of X minus one. And that is what our right hand side looks like. So we successfully match them. So this is answer choice. A yes. Have a nice day.

Verify that each equation is an identity.
tan² α sin² α = tan² α + cos² α - 1

Key Concepts

Trigonometric Identities

Tangent and Sine Functions

Pythagorean Theorem

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