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.

cot² θ(1 + tan² 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 CO C can squared of X multiplied by the quantity of one minus C can't squared X. Our answer choices are answer choice, a negative one divided by sine squared X. Answer choice B one divided by cosine squared X answer choice C one or excuse me, negative one divided by cosine squared X and answer choice D negative one divided by cosine X. All right. This one's fun because there are actually several different ways to solve it. If you take a look at this, what's in parentheses? One minus C can't squared X that looks so close to one of our pythagorean identities. We recall from previous lessons that the tangent squared of X is equal to and the C can't squared X minus one, but we have one minus C can't squared X but we could factor out a negative one. So we take that negative one and that goes in front of our co C can't squared X and that's multiplied by the quantity of factoring out a negative one, we'll now have instead of a negative C can't squared X, that'll be a positive C can't squared X and instead of a positive one, that'll be a minus one. And there we go, that's the same thing as our tangent squared of X. So we can replace the C can squared of X minus one with tangent squared of X. And now that's being multiplied by a negative co C can squared of X. But we are trying to get everything in terms of sign and cosine. Well, that's not too bad. If we take a look at our K can't squared of X, we recall from previous lessons that Kant is the reciprocal function of the sine function. So the Koi can squared of X cos can't squared of X is equal to one divided by the sine squared of X. So we can replace that, then we'll have a negative one divided by sine squared of X. Now that's being multiplied by, we've got the tangent squared of X. But we recall from previous lessons that the tangent squared of X is going to be equal to the sine squared of X divided by the cosine squared of X. And so we can plug that in for a tangent squared of X sine squared X divided by cosine squared X. Now we can multiply straight across in the numerator negative one multiplied by sine squared of X is going to be a negative sine squared X divided by in the denominator, sine squared X multiplied by cosine squared X is sine squared X cosine squared X. But look at that, we've got a sine squared X in both the numerator and the denominator. So those can cancel out. And this simplifies to be in the numerator negative one that's divided by, in the denominator. All that's left is the cosine squared of X. And there we go, we have written it in terms of sign and cosine, we look at our answer choices and this matches with answer choice. C 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.
cot² θ(1 + tan² θ)

Key Concepts

Trigonometric Ratios and Their Definitions

Pythagorean Identity

Algebraic Simplification of Trigonometric Expressions

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