express the function as a trigonometric function of x\ncos 4x\n\nchoose the correct answer below\noa 3 sin x…

express the function as a trigonometric function of x\ncos 4x\n\nchoose the correct answer below\noa 3 sin x - 4 sin 3x\nob. 3 sin x cos 2x - 3 sin 2x cos x\noc. cos 3x - 2 sin x\nod. 8 cos 4x - 8 cos 2x + 1

express the function as a trigonometric function of x\ncos 4x\n\nchoose the correct answer below\noa 3 sin x - 4 sin 3x\nob. 3 sin x cos 2x - 3 sin 2x cos x\noc. cos 3x - 2 sin x\nod. 8 cos 4x - 8 cos 2x + 1

Answer

Explanation:

Step1: Use double - angle formula

We know that (\cos4x=\cos(2\times2x)). By the double - angle formula (\cos2\alpha = 2\cos^{2}\alpha-1), let (\alpha = 2x), then (\cos4x=2\cos^{2}(2x)-1).

Step2: Use double - angle formula again

Since (\cos2x = 2\cos^{2}x - 1), then (\cos^{2}(2x)=(2\cos^{2}x - 1)^{2}=4\cos^{4}x-4\cos^{2}x + 1).

Step3: Substitute back

Substitute (\cos^{2}(2x)=4\cos^{4}x-4\cos^{2}x + 1) into (\cos4x=2\cos^{2}(2x)-1). [ \begin{align*} \cos4x&=2(4\cos^{4}x-4\cos^{2}x + 1)-1\ &=8\cos^{4}x-8\cos^{2}x+2 - 1\ &=8\cos^{4}x-8\cos^{2}x + 1 \end{align*} ]

Answer:

D. (8\cos^{4}x-8\cos^{2}x + 1)