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.

tan x + cot x
---------------------- ; cos x
csc x

Accepted Answer

Hello, everyone. We are gonna write the following trigonometric expression. In terms of the indicated function, we are given the expression open parentheses, tangent of X plus cotangent of X closed parentheses divided by the C can of X. And we want to write this in terms of the sign of X. We are given four answer choices. First, I'm gonna rewrite this in my own handwriting. So again, I have the tangent of X plus the cotangent of X divided by the C can of X from there. I'm gonna rewrite everything in terms of signs and cosigns. So the tangent of X will become the sign of X divided by the cosine of X plus the co tangent of X, which will be the cosine of X divided by the sign of X, all of that divided by one divided by the cosine of X. So I know that looks a little more challenging right now, but it will help us in the long run in my numerator. I'm going to create a common denominator. So my common denominator will be sin X cosine X. I will achieve this by multiplying the numerator and the denominator. The first fraction by the sin of X which will give me the sine squared X divided by the sin of X cosine of X plus my second fraction. I will be multiplying numerator and denominator by the cosine of X. So I will have cosine squared of X divided by the sine of X cosine of X. And I'm gonna leave my denominator as one divided by the cosine of X just for now. Next, I'm gonna combine my numerator into one fraction. So I will have the sine squared X plus cosine squared X divided by the sign of X cosine of X. All of which will be divided by one divided by the cosine of X. I recall that I have an identity that says the sine squared of X plus the cosine squared of X equals one. So this will give me one divided by the sine of X cosine of X divided by one divided by the cosine of X which is the same as saying one divided by the sign of X cosine of X multiplied by the reciprocal of our denominator. So the cosine of X divided by one which is just the cosine of X. And when I cross reduce my cosigns, I am left with one divided by the sine of X. So I have succeeded in my task of putting this in terms of sine and one divided by the sign of X is answer choice. A have a nice day.

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

Key Concepts

Trigonometric Functions

Reciprocal Identities

Simplifying Trigonometric Expressions

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