Simplify each expression. See Example 4.
cos² 2x - sin² ...

Question

Simplify each expression. See Example 4.

cos² 2x - sin² 2x

Accepted Answer

Welcome back. I am so glad you're here. We are asked to simplify the following trigonometric expression. Our expression is the cosine squared of 16 X minus the sine squared of 16 X. Our answer choices are answer choice A, the cosine of 16 X answer choice B, the cosine of eight X answer choice C, the cosine of 64 X and answer choice D, the cosine of 32 X. All right. In order to simplify this, we want to look at our expression and see if it looks similar to any of our identities. And this looks similar to the double angle identity. The one that tells us that the cosine of two multiplied by an angle here angle A equals the cosine squared of that angle minus the sine squared of that angle. And here the right hand side matches up perfectly the cosine squared of an angle minus the sine squared of an angle when our angle A is equal to 16 X. So now we just want to substitute that 16 X anywhere we have an A and then simplify. So on the left hand side, that's the cosine of two multiplied not by A but by 16 X is equal two on the right, the cosine squared of 16 X minus the sine squared of 16 X. We know that our original given expression is identical to the right hand side of this equation for our double angle identity. So we just simplify in the left, we can multiply two multiplied by 16 X. That's going to be 32 X. So this whole thing simplifies to be the cosine of 32 X. We look at our answer choices and that matches with answer choice. D well done. We'll catch you on the next one.

Simplify each expression. See Example 4.
cos² 2x - sin² 2x

Key Concepts

Pythagorean Identity

Double Angle Formulas

Algebraic Manipulation

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