cos-2-x-cos-4x-3-0-x-2pi-

Question Number 167486 by greogoury55 last updated on 18/Mar/22

\cos^{2} x = \cos (\frac{4 x}{3}); 0 ⩽ x ⩽ 2 π

Answered by mr W last updated on 18/Mar/22

\cos 2 x + 1 = 2 \cos \frac{4 x}{3}

\cos \frac{6 x}{3} + 1 = 2 \cos \frac{4 x}{3}

l e t t = \frac{2 x}{3}

\cos 3 t + 1 = 2 \cos 2 t

4 \cos^{3} t - 3 \cos t + 1 = 4 \cos^{2} t - 2

(4 \cos^{2} t - 3) (\cos t - 1) = 0

\Rightarrow \cos t = 1 \Rightarrow t = 2 k π

\Rightarrow \cos t = \pm \frac{\sqrt{3}}{2} \Rightarrow t = k π \pm \frac{π}{6}

\Rightarrow x = 3 k π

\Rightarrow x = \frac{3 k π}{2} \pm \frac{π}{4}

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