Factor each trigonometric expression.
cot⁴ x + 3 cot² x ...

Question

Factor each trigonometric expression.

cot⁴ x + 3 cot² 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 tangent raised to the fourth of X plus six tangent squared X plus five. Our answer choices are answer choice. A, the quantity of tangent squared X plus five multiplied by the quantity of tangent squared X plus one answer choice B, the quantity of tangent squared X minus five multiplied by the quantity of tangent squared X plus one answer. Choice C the quantity of tangent squared X plus five multiplied by the quantity of tangent squared X minus one. And to answer choice D, the quantity of tangent squared X minus five multiplied by the quantity of tangent squared X minus one. All right. So we take a look at our trinomial here and we've got three terms being added that we want to factor and some of them have these tangents. And what we can do to make this a little bit easier to conceptualize is we can look at our middle tangent squared X term and we can use U substitution, we'll set this middle tangent squared X equal to U and then we'll come back to that new substitution at the end to get our tangents back. So starting off with our tangent to the fourth X, that would be tangent squared X squared. So we can use U squared plus six multiplied by instead of tangent squared X, that's going to be U plus five. Now, this is a little bit more recognizable of a trinomial. We set up two empty sets of parentheses in order to do the reverse of the foil process. For our firsts, we ask ourselves what multiplied by what gives us this first term U squared. And that is going to be, you multiplied by you for our lasts. We ask ourselves what multiplied by what gives us this positive five. Keeping in mind that for our outsides plus our insides that will need to add up to a positive six U. Now, if you need to go back to previous lessons and review the factoring process, definitely go for that. But going through this rather quickly our lasts here are going to be a positive five multiplied by a positive one. And you can always foil this out to check our firsts. You multiplied by you is you squared, our outsides U multiplied by one is plus U, our insides five multiplied by U is plus five U and our lasts five multiplied by one is plus five. Combining these middle terms. We'll bring down U squared plus U plus five U is plus six U bring down plus five. This does match. So we did factor it correctly. Now, all we need to do is go back and substitute in for our us, our tangent squared X what the U is equal to. So we'll have the quantity of instead of U that will be tangent squared X plus five, multiplied by the quantity of instead of U that's tangent squared X plus one. And there we have factored our expression completely. We look at our answer choices and this matches with answer choice. A well done. We'll catch you on the next one.

Factor each trigonometric expression.
cot⁴ x + 3 cot² x + 2

Key Concepts

Trigonometric Identities

Factoring Polynomials

Quadratic Form

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