The number of solutions of the equation 4 cos | vedclass

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Question

(C) Given equation: $4 \cos 2 	heta \cos 3 	heta = \sec 	heta$
Since $\sec 	heta = \frac{1}{\cos 	heta}$,we have $4 \cos 2 	heta \cos 3 	heta

Vedclass · Accepted Answer

$6$

Vedclass · Answer

(C) Given equation: $4 \cos 2 	heta \cos 3 	heta = \sec 	heta$
Since $\sec 	heta = \frac{1}{\cos 	heta}$,we have $4 \cos 2 	heta \cos 3 	heta \cos 	heta = 1$.
Using $2 \cos A \cos B = \cos(A+B) + \cos(A-B)$,we get $2 \cos 2 	heta (\cos 4 	heta + \cos 2 	heta) = 1$.
$2 \cos 2 	heta \cos 4 	heta + 2 \cos^2 2 	heta = 1$.
Using $2 \cos^2 2 	heta - 1 = \cos 4 	heta$,we get $2 \cos 2 	heta \cos 4 	heta + \cos 4 	heta = 0$.
$\cos 4 	heta (2 \cos 2 	heta + 1) = 0$.
Case $1$: $\cos 4 	heta = 0 \Rightarrow 4 	heta = (2n + 1) \frac{\pi}{2} \Rightarrow 	heta = (2n + 1) \frac{\pi}{8}$.
For $0 < 	heta < \pi$,$	heta \in \{ \frac{\pi}{8}, \frac{3\pi}{8}, \frac{5\pi}{8}, \frac{7\pi}{8} \}$.
Case $2$: $2 \cos 2 	heta + 1 = 0 \Rightarrow \cos 2 	heta = -\frac{1}{2}$.
$2 	heta = 2n\pi \pm \frac{2\pi}{3} \Rightarrow 	heta = n\pi \pm \frac{\pi}{3}$.
For $0 < 	heta < \pi$,$	heta \in \{ \frac{\pi}{3}, \frac{2\pi}{3} \}$.
Checking $\cos 	heta 
eq 0$: None of these values make $\cos 	heta = 0$.
Total solutions = $4 + 2 = 6$.

The number of solutions of the equation $4 \cos 2 \theta \cdot \cos 3 \theta = \sec \theta$,when $0 < \theta < \pi$,is

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