Write each expression in terms of sine and cosine, and then si...

Question

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.

(sec²θ - 1)/(csc²θ - 1)

Accepted Answer

Hello, everyone. We want to write the following trigonometric expression. In terms of sine and cosine, we'll express our answer in terms of X only. We are given the KC squared of X minus one divided by the cosine squared of X minus one. Our answer choices are a negative cosine squared X divided by sine raised to the fourth power X B negative cosine squared X C negative cosine squared X divided by sine squared XD cosine squared X divided by sine raised to the fourth power X to begin. I'm gonna rewrite our expression. So Kant squared X minus one divided by cosine squared X minus one. Since I want this in terms of sine and cosine, I recall that Koi can has a reciprocal identity with sine. So this means I can rewrite Koi squared X as one divided by sine squared X in my numerator still minus one, all of which is still divided by cosine squared X minus one. I think I want to simplify my numerator. So I'm going to treat minus one as sine squared X divided by sine squared X. So we'll have one divided by sine squared X minus sine squared X divided by sine squared X, all of which will be divided by cosine squared X minus one. So combining what's in my numerator, I will have one minus sine squared X divided by sine squared X, all of which is divided by cosine squared X minus one. I wanna do some substituting using other identities we know. So using a Pythagorean identity, I know that one minus sine squared X is the same as cosine squared X. So I want to replace my numerator of my numerator fraction with cosine squared X now divided by sine squared X in my denominator, I'm going to rewrite this as a negative one. And then in parentheses, one minus cosine squared X. I did this because I recall that one minus cosine squared X is equal to sine squared X. So I'll have cosine squared X divided by sine squared X, all of which is divided by negative one multiplied by in parenthesis sine squared X. So now I think it's time to rewrite this and see if things can reduce or how they can reduce. So I'm gonna rewrite this horizontally so I could see it a little better. So I'm gonna take cosine squared X divided by sine squared X divided by negative one multiplied by the sine squared of X. And I recall multiplying fractions or sorry, dividing fractions and multiplying by the reciprocal. So I'll have cosine squared X divided by sine squared X multiplied by one divided by negative one, which is multiplied by the sine squared of X. So now let's multiply this together. I go straight across the numerator. My numerator is the cosine squared of X straight across my denominator. I get negative one multiplied by sine raised the fourth power of X because squared multiplied by A squared would give us 1/4. And this is very similar to a couple of our answer choices, but they have the negative in the top which is fine. We can move that there and say this is negative cosine squared X divided by sine, raise the fourth power of X. And this is our final answer and it's answer choice. A have a nice day.

Write each expression in terms of sine and cosine, and then simplify the expression so that no quotients appear and all functions are of θ only. See Example 3.
(sec²θ - 1)/(csc²θ - 1)

Key Concepts

Pythagorean Identities

Reciprocal Trigonometric Functions

Algebraic Simplification of Trigonometric Expressions

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