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.

1 cos x
------------------ ﹣ ------------------- ; csc x
1 - cos x 1 + cos x

Accepted Answer

Hello, everyone. We are asked to rate the following trigonometric expression in terms of the indicated function. The expression we are given is one divided by, in parentheses, one minus the sign of X closed parentheses minus the sin of X divided by in parentheses, one plus the sin of X closed parentheses. And we want to rewrite this in terms of C can of X, we are given four answer choices. First, I'm going to rewrite this as fractions. So I can see it a little clearer. So I have one divided by one minus the sign of X minus the sign of X divided by one plus the sign of X. So this makes it a little easier for me to see what I'm working with. So I think the first thing I'm going to do is rationalize my denominators. So I'm gonna multiply both the numerator and the denominator of that first fraction by the conjugate, which is gonna have the same terms as our original denominator, but with a different sign in between. So this would be one plus sine X. Then I'm multiplying both the numerator and the denominator by in that first fraction which would give me one plus the sign of X divided by. And the reason we use conjugates is so we get the difference of squares. So it's gonna be divided by one minus the sine squared of X. And for my second fraction, I'm gonna do the same thing. But I'm multiplying both the numerator and the denominator by one minus the sign of X in here. That will give me the sign of X multiplied by the parentheses. One minus sign of X closed parentheses divided by one minus the sine squared of X. Since both of my fractions have the same denominator, I am going to rewrite this. So they have one single fraction that we're working with. So we will have one plus the sign of X minus the sign of X multiplied by one minus the sign of X in parentheses closed parentheses. And my denominator right now, it's one minus sine squared X. I recall that that is the same as cosine squared X. So I'm gonna substitute that in also, in my next step, I'm distributing negative sign of X into the parentheses, one minus the sign of X. So this will give me one plus the sign of X minus the sign of X plus the sign squared of X divided by the cosine squared of X. I have a positive sign of X and a negative sign of X. So that should cancel out. And I have one plus the sine squared of X divided by the cosine squared of X. Now I'm going to divide this into two fractions. So I have one divided by the cosine squared of X plus the sine squared of X divided by the cosine squared of X. Using my identities, I know that one divided by the cosine squared of X is equivalent to the C can squared of X. And I know that sine squared of X divided by cosine squared of X would give me the tangent squared of X. We're almost there, but we gotta get tangent squared of X in terms of C cans. Well, luckily we have an identity that says that the tangent squared of X is equivalent to the C K squared of X minus one. So when we substitute that in, we have C can't squared X plus the C can't squared X minus one. And this would be the same as saying two C can't squared X minus one. So we got all the way to everything in terms of a sea camp. And that is answer choice C two C can't squared X minus one. Have a nice day.

In Exercises 67–74, rewrite each expression in terms of the given function or functions. 1 cos x ------------------ ﹣ ------------------- ; csc x 1 - cos x 1 + cos x

Key Concepts

Trigonometric Identities

Reciprocal Functions

Rational Expressions

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