In Exercises 67–74, rewrite each expression in terms of the given...

Question

In Exercises 67–74, rewrite each expression in terms of the given function or functions.cos x---------------- + tan x ; cos x1 + sin x

Accepted Answer

Hello, everyone. We are asked to rate the following trigonometric expression. In terms of the indicated function, we are given the sign of X divided by, in parentheses, one plus the cosine of X closed parentheses plus the cotangent of X. And we want to rewrite this in terms of the sign of X. We are given four answer choices. First, I'm just going to rewrite our original expression. The sign of XX divided by one plus the cosine of X plus the cotangent of X. I think I'm going to try to rationalize my denominator meaning I'm gonna multiply top and bottom of that fraction by my conjugate. So I have the same terms but with the opposite sign in between. So I'm gonna multiply top and bottom by one minus the cosine of X. And when I do so I get a numerator, that is the sign of X minus sign of X cosine of X. And my denominator is one minus the cosine squared of X and this will still be plus the cotangent of X. Now, I know that one minus the cosine squared of X is an identity. It is equal to the sine squared of X. So I'm actually gonna factor the sin of X back out of my numerator and go back to having it as the sin of X multiplied by open parentheses, one minus the cosine of X closed parentheses. And that will be divided by the sine squared of X. And it's still gonna be plus the cotangent of X. I rewrote it this way because now I can cancel the sign of X in my numerator with one of the sign of XS in my denominator. So I'm going to do that. And I'm left with one minus the cosine of X divided by the sign of X plus the cotangent of X separating my fraction into two fractions. I will have one divided by the sign of X minus the cosine of X divided by the sine of X plus the cotangent of X. So one divided by the sin of X stays put the cosine of X divided by the sin of X is the cotangent of X and then plus the cotangent of X. So I have a minus cotangent of X and A plus cotangent of X which will cancel out leaving me with one divided by the sign of X. And we have now succeeded in getting that original expression in terms of sign. So our answer choice is a one divided by the sign of X have a nice day.

In Exercises 67–74, rewrite each expression in terms of the given function or functions.cos x---------------- + tan x ; cos x1 + sin x

Key Concepts

Trigonometric Identities

Reciprocal Functions

Simplifying Trigonometric Expressions

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