Question

express the function as a trigonometric function of x
cos 4x

choose the correct answer below
oa 3 sin x - 4 sin 3x
ob. 3 sin x cos 2x - 3 sin 2x cos x
oc. cos 3x - 2 sin x
od. 8 cos 4x - 8 cos 2x + 1

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 \(\cos4x=2\cos^{2}(2x)-1\).

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Answer:

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