Use identities to write each expression in terms of sin θ and ...

Question

Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients appear and all functions are of θ only.

cot(-θ)/sec(-θ)

Accepted Answer

Hello, everyone. We are asked to write the following trigonometric expression. In terms of sine and cosine, we want to express the answer in terms of X only. We are given the tangent of negative X divided by the cosine of negative X. Our answer choices are a negative sine of X divided by cosine of X B negative sine of X divided by cosine squared of X C negative sine squared of X divided by cosine squared of X D negative one divided by cosine of X. First noticing that I have negative X in both of these, I'm gonna look at my odd and even identities. So the tangent of negative X, this can be rewritten as the negative tangent of X using our odd and even identities and the cosine of negative X will be rewritten as the cosine of X. So let's rewrite our original expression using this. So we will have negative tangent of X divided by the cosine of X. Since we want this in terms of sine and cosine, I'm gonna rewrite the negative tangent of X as the negative. And then in parentheses, sine of X divided by cosine of X and divide that by the cosine of X. Now we can keep working with this in a complex fraction. Or I prefer to rewrite this as negative open parentheses sine of X divided by cosine of X closed parentheses divided by the cosine of X. I like to write it horizontally. I see it better. So dividing by the cosine of X means I can multiply it by the reciprocal. So this would equal negative sine of X divided by cosine of X multiplied by the reciprocal of the cosine of X which is one divided by the cosine of X. And if I multiply straight across, I keep the negative out front sin of X multiplied by one is sin of X. The cosine of X multiplied by the cosine of X will give us the cosine squared of X. And that's as simplified as this can be staying in sine and cosine. So our equivalent expression is negative sine of X divided by cosine squared of X. And that's answer. Choice B have a nice day.

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Use identities to write each expression in terms of sin θ and cos θ, and then simplify so that no quotients appear and all functions are of θ only.
cot(-θ)/sec(-θ)