In Exercises 35–38, use the power-reducing formulas to rewrite ea...

Question

In Exercises 35–38, use the power-reducing formulas to rewrite each expression as an equivalent expression that does not contain powers of trigonometric functions greater than 1.

6 sin⁴ x

Accepted Answer

Hello, everyone. We are asked to rate the given expression into a trigonometric function with a power equal to one with the help of power reducing formulas. We are given eight sign raised to the fourth power of X. Our answer choices are a three minus the cosine of four X minus four cosine of two X B two minus the cosine of four X minus four cosine of two X C three minus the cosine of four X plus four cosine of two X D three plus cosine four X minus four cosine of two X. The first thing I want to do is I'm going to rewrite our expression eight multiplied by the sign race, the fourth power of X in terms of sine squared because I know I have formulas that will reduce sine squared. So I'm gonna keep the eight as an eight and put it on the outside of a set of parentheses. And in the parentheses, I will have sine squared X closed parentheses and the whole parentheses is squared. Now I recall that the sine squared of X is equal to one minus the cosine of two X divided by two. So I'm gonna substitute that in to our expression to see if that helps simplify a few things. So I have eight multiplied by, in parentheses, one minus the cosine of two X divided by two closed parentheses squared from there. I'm gonna again, keep the eight on the outside of my parentheses. I recall that if I am raising a fraction to a power I square, both the numerator and the denominator. So that's what I'm going to do. So in my numerator, I have still in parentheses, one minus the cosine of two X closed parentheses squared. In my denominator two squared is four. So I'm going to write that down as four. This helps me because now I can cross reduce my eight and my 44 goes into four, once, goes into eight twice. So I'm now working with two, multiplied by the parentheses, one minus the cosine of two X closed parentheses squared. From here, I am going to again keep my two outside the parentheses for now. And in the parentheses, I'm going to multiply out one minus the cosine of two X, all of it squared. So scoring the first term gives me one, scoring. The last term gives me plus cosine squared of two X. And when I take the product of both terms, which would be negative one cosine of two X and double it, I have negative two cosine of two X closed parentheses. So now I've at least gotten it down from a power of four to I have a power of two from there, I'm going to distribute into the parentheses. So two multiplied by one is two, two, multiplied by the cosine squared of two X would be plus two cosine squared of two X and then two multiplied by negative two cosine of two X would be negative four cosine of two X. OK. So I think I want to use the formula that tells me that the cosine squared of X equals one plus the cosine of two X divided by two. Since we have the cosine squared of two X, we're going to double the angle in this. So we're gonna use the cosine squared of two X will be equal to one plus the cosine of four X divided by two. We're gonna plug that in for the cosine squared of two X in our expression. And when we do, we have two plus two, open parentheses, one plus the cosine of four X divided by two, close parentheses minus four cosine of two X. From here, I have a two multiplied by a fraction with a denominator of two. So I can cross reduce that. So I'm going to, and when I do, I'm left with two plus one plus the cosine of four X minus four cosine of two X. And I wanna see if there's any like terms I can combine one and two will add to three. I cannot combine the cosine of four X and the four cosine of two X. So I have three plus the cosine of four X minus four multiplied by the cosine of two X. And that's as simplified as I can get, but I did do what I needed to and everything is raised to the power of one. So our answer choice D is our answer three plus the cosine of four X minus four multiplied by the cosine of two X. Have a nice day.

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In Exercises 35–38, use the power-reducing formulas to rewrite each expression as an equivalent expression that does not contain powers of trigonometric functions greater than 1. 6 sin⁴ x