Factor each trigonometric expression.
(tan x + cot x)² -...

Question

Factor each trigonometric expression.

(tan x + cot x)² - (tan x - cot x)²

Accepted Answer

Welcome back. I am so glad you're here. We are asked to rewrite the following trigonometric expression by factoring it completely. Our given trigonometric expression is the quantity of the S of X plus the kant of X squared minus the quantity of the S of X minus the Koi can of X squared. Our answer choices are answer choice A two sine squared, X plus two Koi can squared X plus four. Answer choice B two sine squared, X plus two, Koi Can squared X minus four, answer choice C four and answer choice D negative four. All right. So we have two quantities being subtracted from one another. So we're going to want to start by combining those. But actually, if we look at these, there are two quantities that are each squares being subtracted. So this is actually a difference of squares. We recall from previous lessons that a difference of squares would be when we have A squared minus B squared. And then that equals to when it's factored out the quantity of A plus B multiplied by the quantity of A minus B. So what's our A and what's our B? Well, A is the first thing being squared. So A is actually going to be the quantity of sine X plus C can't X. That's the first thing squared and B is the second quantity that's squared. So B here is going to equal sine X minus KX. And so if we want to factor those, we'll plug in for our A and our B into what the A squared minus B squared is equal to. So that is the quantity of A plus B A is sine X plus C can't X plus our B which is sine X minus kant X and that's multiplied by the quantity of A minus B. So A again, sine X plus K can't X minus our B. I'm going to put this in parentheses. So we don't forget to distribute that negative to both parts of our B. So B is sine X minus Kant X and so now we can do some simplification here. We factor it but we can simplify this. So looking at that first long quantity sine X plus CC can't X plus sine X minus KC. Can't X those kant XS cancel out and sine X plus sine X is two sine X. Now that's being multiplied by the second quantity. So let's look in there and simplify that we've got sine X plus CC can X minus both sin X and negative COC can X. So the sine X minus sine X is going to cancel out and then this is going to be Kant X minus negative Kant X which is plus K can't X. So Kant X plus Kant X is two Kant X. Now we can multiply these together where we've got two sine X multiplied by two, Koi can't X, that's going to be two multiply by two is four. And then we have sine X multiplied by Kant X. But we recall from previous lessons that sign and KC can't are reciprocal functions. So if we have the sign of X, that's equal to one divided by the cos of X. And so we can use substitution for the sine of X in here. And so this would be four multiplied not by sin of X anymore, but by one divided by KC can't X multiplied by KC can't X and now those KC can XS will cancel each other out. It's just four multiplied by one which simplifies to be four. That whole thing simplifies to be four. We look at our answer choices and this matches with answer choice. C well done. We'll catch you on the next one.

Factor each trigonometric expression.
(tan x + cot x)² - (tan x - cot x)²

Key Concepts

Trigonometric Identities

Difference of Squares

Factoring Techniques

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