The number of solutions of the equation 4 cos | vedclass

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(C) We have,$4 \cos 2 	heta \cdot \cos 3 	heta = \sec 	heta$ for $0 < 	heta < \pi$. Multiplying by $\cos 	heta$ (since $\cos 	heta 
eq 0$ for $

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$6$

Vedclass · Answer

(C) We have,$4 \cos 2 	heta \cdot \cos 3 	heta = \sec 	heta$ for $0 Multiplying by $\cos 	heta$ (since $\cos 	heta 
eq 0$ for $\sec 	heta$ to be defined),we get $4 \cos 2 	heta \cdot \cos 3 	heta \cdot \cos 	heta = 1$. Using $2 \cos A \cos B = \cos(A+B) + \cos(A-B)$,we have $2 \cos 2 	heta (\cos 4 	heta + \cos 2 	heta) = 1$. $2 \cos 4 	heta \cos 2 	heta + 2 \cos^2 2 	heta = 1$. Using $2 \cos^2 2 	heta - 1 = \cos 4 	heta$,we get $2 \cos 4 	heta \cos 2 	heta + \cos 4 	heta + 1 = 1$. $\cos 4 	heta (2 \cos 2 	heta + 1) = 0$. Case $1$: $\cos 4 	heta = 0$ $\Rightarrow 4 	heta = \frac{\pi}{2}, \frac{3 \pi}{2}, \frac{5 \pi}{2}, \frac{7 \pi}{2}$ $\Rightarrow 	heta = \frac{\pi}{8}, \frac{3 \pi}{8}, \frac{5 \pi}{8}, \frac{7 \pi}{8}$. Case $2$: $2 \cos 2 	heta + 1 = 0$ $\Rightarrow \cos 2 	heta = -\frac{1}{2}$ $\Rightarrow 2 	heta = \frac{2 \pi}{3}, \frac{4 \pi}{3}$ $\Rightarrow 	heta = \frac{\pi}{3}, \frac{2 \pi}{3}$. All these $6$ values lie in the interval $(0, \pi)$. Thus,the number of solutions is $6$.

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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