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 + cot(-θ)/cot(-θ)

Accepted Answer

Welcome back. I am so glad you're here. We are asked to write the following trigonometric expression in terms of sine and cosine express your answer in terms of X. Only our given trigonometric expression is one plus the tangent of negative X. And that's all in the numerator, that's all divided by the cotangent of negative X. Our answer choices are answer choice a negative. And then in the numerator, we have sine X multiplied by cosine X plus sine squared X. And that's all divided by cosine squared X answer choice B negative. And then again, we have a fraction in the numerator sine X multiplied by cosine X minus sine squared X. That's all divided by cosine squared X. Answer choice C negative. Again, a fraction in the numerator sine X multiplied by cosine X minus sine X all divided by the cosine squared of X. And answer choice D negative. Again, a fraction in the numerator sine X multiplied by cosine X minus sine squared X. And that's all divided by the cosine of X. All right. So a few things to note here, we see that we a tangent of negative X and a cotangent of negative X. And we recall from previous lessons that tangent and cotangent are both odd functions, which means that for the tangent and the cotangent of a negative X, that's going to be equal to a negative tangent and cotangent of X. So we can rewrite this, this is equal to one instead of plus tangent of negative X, that's equal to one minus tangent of straight up X that's divided by instead of writing the cotangent of negative X, that's the same as the negative cotangent of X. All right. Now, we want to get this in terms of sign and cosine and we can get tangent and cotangent to sign and cosine pretty easily. But it can look a little bit messy with the fractions and the fractions. So I'm going to rewrite this as for we know that the tangent of X, we recall from previous lessons that that is equal to the sine of X divided by the cosine of X. So rewriting just that numerator as its own fraction, it'll be one minus sine X divided by cosine X. Now that whole thing divided by, with a division simple. Now, I'm going to rewrite the cotangent of X. We recall from previous lessons that that's the reciprocal of the tangent that's going to be cosine of X divided by sine of X. So rewriting that remembering that the cotangent here is negative. So it'll be a negative in the numerator cosine X divided by the sine of X. Now, a couple things we want to do for our first thing here, one minus sine X divided by cosine X. We want to combine those to be one fraction. And so we'll have to have a common denominator. We'll multiply that one by cosine X divided by cosine X. And then when we do that, we have cosine X minus sine X in the numerator over our common denominator cosine X. Now, instead of dividing by negative cosine X divided by sine X, let's multiply by the reciprocal which will be a negative sine X divided by cosine X. And so now we just need to distribute that negative sine X in the numerator to our cosine X minus sine X. So negative sine X multiplied by cosine X will be a negative sine X, cosine X and the negative sign X multiplied by negative sign X is going to be plus sign squared X. And then that is all divided by, in the denominator. Then we've got cosine X multiplied by cosine X. So that's going to be cosine squared X. And that is it in the numerator, we can factor it out so that it looks like our answer choices, we can factor out a negative in front. And so then we'll have in the numerator sine X cosine X. And then instead of plus sine squared X, we factored out the negative. So it'll be minus sine squared X all divided by cosine squared X. And there it is, we look at our answer choices and this matches with answer choice. B well done. We catch you on the next one.

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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 + cot(-θ)/cot(-θ)