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

Question

In Exercises 67–74, rewrite each expression in terms of the given function or functions. (sec x + csc x) (sin x + cos x) - 2 - cot x; tan x

Accepted Answer

Hello, everyone. We are asked to write the following trigonometric expression. In terms of the indicated function we are given in parentheses. The co C can of X minus the C can of X multiplied by the parentheses, cosine of X plus sin of X closed parentheses plus two plus the cotangent of X. And we want to rewrite this in terms of the tangent of X, we have four answer choices. So first I am going to rewrite the given expression and I'm going to change the cos of X into one divided by the sign of X. And that will be subtracted by the C K of X, which is the same as one divided by the cosine of X. And that's multiplied by the cosine of X plus the sin of X closed parentheses plus two plus the cotangent of X. Since I want to get everything in terms of tangent, I'm gonna see what I can do to get rid of all these signs and cosigns. So in the first set of parentheses, I'm gonna make a common denominator. So for the first fraction, one divided by the sin of X, I'm multiplying top and bottom or numerator and denominator by the cosine of X. So I get the fraction cosine X divided by sine of X cosine of X in the second fraction. That first set of parentheses, I'm multiplying numerator and denominator by the sin of X which will give me the sin of X divided by the sin of X cosine of X closed parentheses. And that is still multiplied by the parentheses, cosine of X plus sin of X closed parentheses plus two plus the cotangent of X. Next, I'm rewriting my fraction in my first set of parentheses as a single fraction. So I will have the cosine of X minus the sign of X divided by the sign of X cosine of X closed parentheses multiplied by the cosine of X plus the sin of X closed parentheses plus two plus the cotangent of X. Now, I notice that I can multiply together my two sets of parentheses. And when I do so, my numerators are going to be the product of a sum and a difference. So it's gonna be the difference of squares. So I'll have cosine squared X minus sine squared X divided by the sign of X cosine of X. And that will be plus two plus the cotangent of X. Now, I know it seems like we just put that together into one fraction, but I'm gonna break my parentheses apart into two fractions. So I will have the cosine squared of X divided by the sin of X cosine of X minus the sine squared of X divided by the sin of X cosine of X plus two plus the cotangent of X. So we did this because now I can reduce these fractions. I can reduce one of the cosine XS in the numerator with the cosine X in the denominator of the first fraction. Leaving me with the cosine of X divided by the sine of X In the second fraction, I can do a similar thing with the signs, I can reduce one of the signs of X with a sign of X in the denominator. Leaving me with the sign of X divided by the cosine of X plus two plus the cotangent of X cosine of X divided by the sin of X is the cotangent of X minus the sin of X divided by the cosine of X is the tangent of X plus two plus the cotangent of X. We're almost there. I'm going to combine my two cotangents together. So I have two cotangent X minus tangent X plus two and one more step. And everything will be in terms of tangents. And that step is to change the cotangent of X into two divided by the tangent of X. So we will have two divided by the tangent of X minus the tangent of X plus two. And we will have accomplished our task of rewriting this in terms of tangent. So we have two divided by the tangent of X minus the tangent of X plus two. And this matches with answer choice A, have a nice day.

In Exercises 67–74, rewrite each expression in terms of the given function or functions. (sec x + csc x) (sin x + cos x) - 2 - cot x; tan x

Key Concepts

Reciprocal Trigonometric Functions

Algebraic Manipulation of Trigonometric Expressions

Pythagorean and Quotient Identities

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