In Exercises 39–42, use double- and half-angle formulas to find t...

Question

In Exercises 39–42, use double- and half-angle formulas to find the exact value of each expression.

cos² 15° - sin² 15°

Accepted Answer

Hello, everyone. We are asked to determine the exact value of the following expression by writing it as the cosine of a double angle. The expression we are given is cosine squared of 67. degrees minus sine squared of 67.5 degrees. Our answer choices are a negative one half B negative radical two divided by two C negative radical three divided by two D one. So the instructions tell us, we want this to be written as the cosine of a double angle. So the cosine of a double angle identity says that the cosine of two times the equals the cosine squared of theta minus the sine squared of theta. Well, what we have right now is the cosine squared of 67.5 degrees minus the sine squared of 67. degrees. So do we think that theta should be 67.5 degrees? It looks like it would match up. So if we say that theta in this example equals 67.5 degrees, if we substitute that into the double angle piece, the very first part where it says that the cosine of two multiplied by the. So we'd have the cosine of two multiplied by 67.5 degrees, which would be the same as saying the cosine of degrees, which if you refer to a unit circle, you will find the exact value of this is negative radical two divided by two. So the exact value of the cosine squared of 67.5 degrees minus sine squared of 67.5 degrees is the cosine of degrees, which is negative radical two divided by two. And that's answer choice B have a nice day.

In Exercises 39–42, use double- and half-angle formulas to find the exact value of each expression. cos² 15° - sin² 15°

Key Concepts

Double-Angle Formulas

Half-Angle Formulas

Pythagorean Identity

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