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 cos θ tan θ

Accepted Answer

Welcome back. I am so glad you're here. We are asked to write the following trigonometric expression in terms of sine and cosine express your answer in terms of X. Only our given trigonometric expression is the tangent of X multiplied by the sign of X multiplied by the C can of X. Our answer choices are answer choice. A sine squared X divided by cosine squared X, answer choice B sine squared X minus cosine squared X, answer choice C sine squared X and answer choice D cosine squared X. All right. So one of our terms is already in terms of sign and cosine, that's that sign of X. But we've got to do a little bit of work for the tangent of X and the C of X. So let's work on the tangent of X. First, we recall from previous lessons that the tangent of X is equal to the sine of X divided by the cosine of X. So, hey, that's in terms of sine and cosine. Now let's replace tangent of X with sine of X divided by cosine of X. And then we can bring down that multiplied by the sine of X and then we need to figure out the C can of X. Well, we recall from previous lessons that the C can't of X is the reciprocal function of the cosine of X. So the C can of X is equal to one divided by the cosine of X. And that is in terms of sine and cosine. So we can replace the C can of X with one divided by the cosine of X. Now, we can multiply straight across in our numerators, sine of X multiplied by sine of X multiplied by one is sine squared X that's divided by multiplying our cross, our denominators cosine of X multiplied by the cosine of X is the cosine squared X. And that's it. We have simplified this and written it in terms of sign and cosign, we look at our answer choices and that matches with answer choice. A well done. We'll catch you on the next one.

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 cos θ tan θ

Key Concepts

Reciprocal and Quotient Identities

Simplification of Trigonometric Expressions

Function Domain and Variable Consistency

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