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.

[1 - sin²(-θ)]/[1 + cot²(-θ)]

Accepted Answer

Welcome back. I am so glad you're here. We're asked to write the following trigonometric expression in terms of sine and cosine express your answer in terms of X, only our given expression is in the numerator, we have one plus sine squared of negative X. And that's divided by, in the denominator, one minus the tangent squared of negative X. And our answer choices are answer choice A this is a fraction in the numerator. We have the quantity of one plus sine squared X multiplied by cosine squared X. That's all divided by cosine squared X minus sine squared X. Answer choice B is also a fraction in the numerator. We have one minus s signed to the fourth X. That's all divided by cosine squared X. Answer. Choice C is a fraction but it's also a fraction within a fraction. The numerator is a fraction. The numerator's numerator is one plus sine squared X and that's divided by the numerator's denominator which is cosine squared X minus sine squared X and that whole numerator is divided by the cosine of X and answer choice D is just one. All right. So a couple things to notice here. We noticed that both the sign and the tangent are being taken of a negative X and both sine and tangent are odd functions, which means that when we have the sign of a negative value, so a negative X that's going to be equal to a negative sine of now a positive X and the same thing with the tangent. However, we've got a sine squared and a tangent squared. So if we took a negative sign of X multiplied by a negative sign of X, that's going to get us right back to a positive sign of X. So rewriting this numerator, it's gonna be one plus sine squared of now positive X that's divided by. Now in the denominator, we have a negative multiplied by a negative tangent of X multiplied by a negative tangent of X. And there's three negatives there, two of them cancel out. And so we're left with pretty much what we had at the start one minus tangent squared. But now instead of negative X, that's a positive X. So that was step one. The next thing we're going to do is we want to get that tangent into terms of sign and cosine. So I am going to start off here. We know that the tangent squared of a value we'll say X is equal to in the numerator that's going to be the sine squared of X and that's divided by, in the denominator cosine squared of X. So let's rewrite this. Our numerator is one plus sine squared of X. And then that whole thing is divided by. And I'm going to write it with a division symbol one minus. And instead of writing tangent squared of X, I'm going to write sine squared X divided by cosine squared X. And now we can do a little bit of work. That second part, what's being divided the one minus sine squared X minus or divided by cosine squared X, we want to combine that into one fraction. And in order to do that, we need to have a common denominator. So we're going to multiply that one by cosine squared X divided by cosine squared X. So then we'd have a common denominator. So we can combine our numerators, we'd have cosine squared X minus sine squared X all divided by cosine squared X. And then that whole thing is still from the, we have one plus sine squared X divided by that whole thing. Now, we can, because we're dividing fractions, we can rewrite this by multiplying by the reciprocal of the second fraction. So we'll have one plus the sine squared of X multiplied by the reciprocal of cosine squared X minus sine squared X all divided by cosine X which is now in the numerator, we'll have cosine squared X divided by, in the denominator, cosine squared X minus sine squared X. And our answer choices actually keep it very similar to this multiplying straight across in our numerators, they don't distribute it. They've got cosine squared X multiplied by the quantity of one plus sine squared X. Actually, they put that quantity first. But that's the same thing all divided by our denominator is cosine squared X minus sine squared X. And that one is going to match mostly if you imagine that I put one plus sine squared X first and then multiply that by cosine squared X, it's going to match with answer choice. A well done. We'll catch you on the next one.

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.
[1 - sin²(-θ)]/[1 + cot²(-θ)]

Key Concepts

Trigonometric Identities

Cotangent Function

Negative Angle Identities

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