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 = 1 - 2 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 the equation is an identity, we want 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 the Pythagorean identity for cosine of two X. So let's take a look at the given equation. We are given five multiplied by cosine of two X is equal to negative five plus 10 multiplied by sine squared of X. Let's go ahead and list down the Pythagorean identities for cosine of two X cosine of two X is equivalent to three different Pythagorean identities. The first one is cosine squared X minus sine squared X. The second identity is two multiplied by cosine squared X minus one. And the third identity is one minus two multiplied by sine squared of X. So what we wanna do is you want to pick one of the three options that closely resembles the right hand side of the given equation. Notice that the right hand side of the given equation has a constant number in its first term and a sine squared value in its second term, the one option from the three that closely resembles the right hand side is going to be option 31 minus two multiplied by sine squared X. So if we rewrite cosine two X with this quantity, we can rewrite the left hand side of the equation as five multiplied by one minus two multiplied by sine squared X. And this is going to equal to negative five plus 10 multiplied by sine squared of X. In order to simplify the left hand side, we're going to distribute five into one minus two S scored X. This will allow us to rewrite the left hand side as five minus 10 multiplied by sine squared of X equal to negative five plus 10 multiplied by sine squared of X. Now, at this point, there's nothing further that we can do to rewrite the left hand side to equal to the right hand side. Furthermore, the right hand side is the negative opposite of the left hand side, which means that this is not an identity. Therefore, this answer to this problem is going to be B 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 (Hint: cos 2x = cos(x + x).)
cos 2x = 1 - 2 sin² x

Key Concepts

Trigonometric Identities

Double Angle Formulas

Verification of Identities

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