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.

csc θ - sin θ

Accepted Answer

Hello, everyone. We are asked to write the following trigonometric expression. In terms of sine and cosine. We want to express our answer in terms of X only. We are given the cotangent of X minus the C of X. Our answer choices are a cosine squared X minus sine X divided by sine X cosine X B cosine squared X plus sine X divided by sine X cosine X C cosine squared X minus sine X divided by cosine X D cosine squared X minus sine X divided by sine X. First, I'm gonna rewrite my original expression. So cotangent X minus the C of X. And then thinking through my identities to rewrite these as sine and cosine, cotangent of X will be the same as the cosine of X divided by the sine of X. And we're gonna subtract the C cant of X which through reciprocal identity is one divided by the cosine of X. Now that we have two fractions, we must make them have the co a common denominator. Our common denominator here will be sin X cosine X. So I'm going to set up my next two fractions with the common denominator of Sinex cosina. So for my first fraction to get from Sinex to Sinex cosina, I must multiply both the numerator and the denominator by the cosine of X. And my new numerator will be cosine squared X for the second fraction to get from the cosine of X to sine X cosine of X. I must multiply the numerator and denominator by the sine of X. And when I do so, I will have SX as my new numerator combining this into one fraction with the common denominator of sine X cosine X. My numerator will be cosine squared X minus sine X. And there's nothing else there that looks like it could be simplified and still remain in terms of sign and cosine. So our answer is cosine squared X minus sine X divided by sine X cosine X which is 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.
csc θ - sin θ

Key Concepts

Cosecant Function

Sine Function

Simplification of Trigonometric Expressions

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