Simplify each expression. See Example 4.
cos² π/8 - 1/2

Question

Simplify each expression. See Example 4.

cos² π/8 - 1/2

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 pi divided by 12 minus one half. Our answer choices are answer choice. A 1/4 answer choice, B one half, answer choice C square root three divided by two and answer choice D square root three divided by four. All right. So we take a look at our trigonometric expression and ask ourselves if it looks like any of our trigonometric identities in order to simplify it. And this one looks a bit like one of our double angle identities specifically that the cosine of two times an angle, I'll write capital A is equal to two multiplied by the cosine squared of that same angle. A minus one. It's very similar. We've got the cosine squared of an angle minus something, but it's minus one half in our given expression and it's minus one in our double angle identity. But you know what? We can work a little bit of magic here to get it to look even more like our trigonometric expression. If we divide everything on both sides buy two, what we'll have on the left, we have the cosine of two multiplied by the angle A all divided by two equals on the right two divided by two cancels out. So that's going to be the cosine squared of angle A and then it's minus one half. So that looks pretty much just like our given expression. So what we can do is we can set our angle A equal to the pi divided by 12 in our given expression. So let's fill that in. We have the cosine of two multiplied by angle A is pi divided by 12, all divided by two equals our given cosine squared of pi divided by 12 minus one half. So now we can just work to simplify the left hand side of the equation because when we multiply two by pi divided by 12, that's going to be the cosine of pi divided by six, not 16 6, that's still divided by two. But hey, the cosine of pi divided by six, we remember that from previous lessons that gives us an exact value pi divided by six is on the unit circle. And recalling from those same previous lessons that the cosine of an angle is equal to the X value where that angle hits the unit circle. And here the X value on the unit circle is going to be square at three divided by two. So here in the numerator, the cosine of pi divided by six is equal to the square root of three divided by two, that's still divided by two, which is going to equal the square root of three divided by four. And that's our simplified trigonometric expression. We look at our answer choices and this matches with answer choice. D well done. We'll catch you on the next one.

Simplify each expression. See Example 4.
cos² π/8 - 1/2

Key Concepts

Trigonometric Identities

Cosine Function

Half-Angle Formulas

Watch next

Simplify each expression. See Example 4.cos² π/8 - 1/2

Key Concepts

Trigonometric Identities

Cosine Function

Half-Angle Formulas

Watch next

Simplify each expression. See Example 4.
cos² π/8 - 1/2