Write each expression in terms of sine and cosine, and then si...

Question

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

-sec² (-θ) + sin² (-θ) + cos² (-θ)

Accepted Answer

Hello, everyone. We are asked to write the following trigonometric expression in terms of sine and cosine. We'll express our answer in terms of X. We are given negative kant squared of negative X minus cosine squared of X minus sine squared of negative X. Our answer choices are a, the parenthesis is negative one minus sine squared X divided by sine squared X B. The parentheses one minus sine squared X all of which is divided by sine squared X C. The parenthesis negative one minus sine squared X divided by the cosine squared of X D. The parentheses negative one minus sine squared X, all of which is divided by sine X. OK. So I wanna rewrite this just so I can see it a little better. So I have the negative co C can't squared of negative X minus the cosine squared of X minus the sine squared of negative X. All right. First thing I want to deal with is these negative XS. So I recall that the sign of negative X is equal to the negative sign of X and that the KC can of negative X is equal to the negative KC can't of X. So let's see how that could help us. So I'm gonna put a negative sign and open a set of parentheses because if I'm rewriting the KC can't squared of negative X, this is like saying I'm doing the negative co C can of X, all of it is squared and I'll show you what, why that makes a difference in the next step. But I'm gonna do the same with the sine. So minus cosine squared X, that'll stay as is and then minus, I'm going to put it again in parentheses and rewrite the S squared of negative X as negative sine of X in parentheses squared. So keeping that negative out front but squaring my parentheses, negative KC can't X squared and negative times a negative is a positive. So now this will be KC can't squared X. So I have negative CC can't squared X minus cosine squared X. And the same thing happens here with the negative sign of X squared. When I square something it becomes a positive, there was already a minus sign in front of it. So that one stays put. So at this step, this is simplified to the negative KC can't squared of X minus the cosine squared of X minus the sine squared of X. From here, I notice I have cosine squared X and sine squared X. And I recall that we have an identity that states that the sine squared of X plus the cosine squared of X equals one. So I think I can rearrange this to make that identity work here. So I'm leaving the negative KC can't squared X by itself for the moment, I'm going to factor out a negative open a set of parentheses and rewrite this as the sine squared of X plus the cosine squared of X because I took the negative sign away from both and put it outside the parentheses. I can now use that identity. So I'll have the negative KC can squared of X minus one. All right. So next piece I need to consider what Ky can't is. So thinking about my reciprocal identities, I know that cos can of X is equal to one divided by the sine of X because we have a reciprocal identity. So let's use that to our advantage. So let's rewrite this as negative one divided by the sign squared of X and then minus one, we want to simplify this into one single fraction because that's what all of our answer choices look like. So I'm gonna rewrite our minus one. So we have negative one divided by the sine squared of X minus. I'm rewriting this as sine squared X divided by sine squared X. Now that we have a common denominator, I can write it together. So I'll have negative one minus sine squared X divided by sin squared X. And that looks as simplified as this can get. And so negative one minus sine squared X, all of which is divided by sine squared X is answer choice a have a nice day.

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.
-sec² (-θ) + sin² (-θ) + cos² (-θ)

Key Concepts

Trigonometric Identities

Even and Odd Functions

Simplification of Trigonometric Expressions

Watch next