Verify that each equation is an identity.
(sin² θ)/cos θ...

Question

Verify that each equation is an identity.

(sin² θ)/cos θ = sec θ - cos θ

Accepted Answer

Hello. Today we are going to be proving whether the given equation is an identity or not. The equation given to us is cotangent squared of X divided by coin of X is equal to coin of X minus sine of X. Now, in order to start this process, we want to show that the left hand side of the equation will equal to the right hand side of the equation to make things a little easier to look at. Let's go ahead and rewrite the left hand side of the equation to be a product, we can rewrite the left hand side to be cotangent squared of X multiplied by the quantity one divided by coin of X. There are two trigonometric identities that we can use to help us. In this case. The first trigonometric identity is that sin of X is equal to one divided by cos seeking of X. We can use this trigonometric identity to rewrite one divided by co seeking of X as sin of X. Furthermore, we do have a trigonometric identity that involves cotangent squared of X. That identity is given to us as one plus cote and squared of X is equal to co seek and squared of X if we were to solve for co tangent squared in the identity, by subtracting one to both sides of the equation, we can rewrite the identity to be co tangent squared of X is equal to COK and squared of X minus one. We can use this to rewrite cotangent squared of X to be co seeking squared of X minus one, applying these substitutions to the left hand side while that was to rewrite it as the quantity cok and squared of X minus one multiplied by sine of X. If we were to distribute sin of X into the quantity, we can rewrite the left hand side of the equation to be coque and squared of X multiplied by sin of X minus sine of X. Now, this is becoming very similar to the right hand side of the equation. We have the minus sine X term. But now we need to figure out a way to reduce kin squared of X multiplied by sin of X to just be coin of X. Well, in order to do this, we can use the sign of X identity one more time. If we were to rewrite the sine of X as one divided by co seeking of X, we can rewrite the left hand side to be co seeking squared of X multiplied by one divided by co seeking of X minus sign of X. This will allow us to reduce Kosekin squared of X to just be coin of X. And the left hand side of the equation will be coin of X minus sine of X, which is equal to the right hand side of the given equation proving that this is an identity. So 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 θ)/cos θ = sec θ - cos θ

Key Concepts

Trigonometric Identities

Reciprocal Functions

Algebraic Manipulation

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