Perform each indicated operation and simplify the result so th...

Question

Perform each indicated operation and simplify the result so that there are no quotients.

cos β(sec β + csc β)

Accepted Answer

Welcome back. I am so glad you're here. We are asked to simplify the given trigonometric expression. Our given expression is the cotangent of X multiplied by the quantity of the tangent of X plus. The sign of X. Our answer choices are answer choice. A cosine of X answer choice BC can't of X answer choice C one plus cosine of X and answer choice D one plus sine of X. All right. So to simplify this, we're going to begin by distributing our cotangent of X to what's inside the parentheses, the tangent of X and the sign of X. So we'll have cotangent of X multiplied by tangent of X plus cotangent of X multiplied by sign of X. And now we just want to think about other ways to express each of these trigonometric functions so that we can maybe get some things to cancel out here. For the first one cotangent of X multiplied by tangent of X. We can recall from previous lessons that the cotangent of X can also be expressed as the reciprocal function of the tangent function. So one divided by the tangent of X. So if we substitute that in for this first cotangent of X, we'll have one divided by the tangent of X multiplied by the tangent of X. And there we can see our tangent of X's cancel out and that will just be one now plus this cotangent of X multiplied by the sine of X. Here, we can think of something else for our cotangent of X. We recall from previous lessons, we can also express cotangent of X as cosine of X divided by sine of X. So we can substitute that. And for the second cotangent of X, we'll have cosine of X divided by sine of X multiplied by sine of X. And for this one, now our sine X functions cancel out and this will be plus just cosine of X. So this whole expression simplifies to be one plus cosine X. We look at our answer choices and this matches with answer choice. C well done. We'll catch you on the next one.

Perform each indicated operation and simplify the result so that there are no quotients.
cos β(sec β + csc β)

Key Concepts

Trigonometric Functions

Reciprocal Identities

Simplification of Trigonometric Expressions

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