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 x/sec x + sin x/csc x

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 C can't of X divided by the cosine of X minus the tangent of X divided by the cotangent of X. Our answer choices are answer choice, a negative one answer, choice B zero, answer choice C one and answer choice D two C can't X. All right. So we take a look at this and in order to simplify it first, we want to see what the relationships are between these various trigonometric functions. And within each of our fractions, the numerator and the denominator are reciprocal functions of each other. So how does that work out? Well, if we have the C can of X divided by the cosine of X, we know that for example, the cosine of X equals one divided by the sea can't of X. And for the tangent of X divided by the cotangent of X, we can look at that cotangent of X in the denominator and that's equal to one divided by the tangent of X. Those are their reciprocal functions. So if we plugged in what those are both equal to, how would that simplify? So we'll start off with our C cant of X divided by and instead of the cosine of X, that's going to be divided by one divided by the C can of X then minus and for the tangent of X divided by cotangent of X, that's now the tangent of X divided by, instead of cotangent of X, it's one divided by a tangent of X. So those will simplify, if we think about it for a moment, we'll go off to the side. If we have the C can't of X, we write this with a division symbol divided by one divided by the C can't of X. We recall that that is the same as multiplying by the reciprocal of the second fraction. So this will be the C can't of X multiplied by the C can't of X. And when that's multiplied by itself, that's going to be C can't squared X. So our first fraction is going to simplify to be C can't squared X. And then we have the same thing going on with the tangent of X divided by one divided by the tangent of X. So this will be minus tangent squared X. Now we want to recall from previous lessons and from the textbook, whether this is any sort of fundamental trigonometric identity. And this is, we recall from previous lessons, we have the identity where the tangent squared of an angle will say theta plus one is equal to the C squared of that same angle theta. Now, it doesn't look exactly the same, but it's very similar. If we were to take both sides and subtract tangent squared of, well, we'll stick with theta and subtracting tangent squared, theta from both sides we'd have on the left. That's a positive one equals on the right C can't squared theta minus tangent squared, theta. And if our theta is equal to X, then that right hand side is exactly the same as what we've got. C can't squared X minus tangent squared X. So that thing is equal to our expression is equal to one. So our whole expression simplifies to be one. We look at our answer choices and that matches with answer choice. See, 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 x/sec x + sin x/csc x

Key Concepts

Trigonometric Functions

Reciprocal Identities

Simplifying Trigonometric Expressions

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