Verify that each equation is an identity (Hint: cos 2x = cos(x...

Question

Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)

cos 2x = cos² x - sin² x

Accepted Answer

Hello. Today we're going to be verifying whether the given equation is an identity. Now, in order to verify if this 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. In order to do this, we're going to be using different Pythagorean identities. So the equation given to us is cosine of two X is equal to negative cosine squared of X plus sine squared of X. Now, on the left hand side, we are given cosine two X. So let's go ahead and list down the Pythagorean identity for cosine of two X. Now cosine of two X has three different identities. The first identity is that it is equal to cosine squared X minus sine squared X. The second identity it's equal to is two cosine squared of X minus one. And the third identity that it is equal to is one minus two sine squared of X. So when you are rewriting cosine two of X, you can set it equal to any one of the three options. Now, in this case, we want to pick the option that closely matches the right hand side of this equation. The equation involves cosine squared X and sine squared X. And the one option that involves both of these terms is the first option. So we can rewrite the left hand side of the equation as cosine squared X minus sine squared X. So if we do this, we have cosine squared of X minus sine squared of X is equal to negative cosine squared of X plus sine squared of X. Now, currently, there's nothing further that we can do to the left hand side to make it equal to the right hand side. Since the right hand side is the negative opposite of the left hand side, what this means is that this equation is not an identity. And this means that the answer to the problem is going to be B So I hope this video helps you in understanding how to verify this identity. And I'll go ahead and see you all in the next video.

Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)
cos 2x = cos² x - sin² x

Key Concepts

Trigonometric Identities

Double Angle Formulas

Cosine Function Properties

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