The number of solutions of the equation
$\cos 2 \theta \cos \frac{\theta}{2}+\cos \frac{5 \theta}{2}=2 \cos ^3 \frac{5 \theta}{2}$
in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ is :

Question

The number of solutions of the equation $\cos 2 	heta \cos \frac{	heta}{2}+\cos \frac{5 	heta}{2}=2 \cos ^3 \frac{5 	heta}{2}$ in $\left[-\frac{\pi}{2}, \frac{\pi}{2}ight]$ is :

Aicharya · Accepted Answer

$\begin{aligned} & \cos 2 	heta \cos \frac{	heta}{2}+\cos \frac{5 	heta}{2}=2 \cos ^3 \frac{5 	heta}{2} \ & \frac{1}{2}\left(2 \cos 2 	heta \cos \frac{	heta}{2}ight)+\cos \frac{50}{2} \ & =\frac{1}{2}\left(\cos \frac{15 	heta}{2}+3 \cos \frac{5 	heta}{2}ight) \end{aligned}$ or solving $\begin{aligned} & \cos \frac{3 	heta}{2}=\cos \frac{15 	heta}{2} \ & \cos \frac{15 	heta}{2}-\cos \frac{3 	heta}{2}=0 \ & 2 \sin 30 \sin \frac{9 	heta}{2}=0 \end{aligned}$ $3 	heta=\mathrm{n} \pi 	ext { or } \frac{9 	heta}{2}=\mathrm{m} \pi$ $\begin{aligned} & 	heta=\frac{\mathrm{n} \pi}{3} \quad 	heta=\frac{2 \mathrm{~m} \pi}{9} \ & 	heta=\left\{-\frac{\pi}{2}, \frac{\pi}{3}, 0ight\} \ & 	heta=\left\{-\frac{4 \pi}{9}, \frac{-2 \pi}{9}, \frac{4 \pi}{9}, \frac{2 \pi}{9}ight\} \end{aligned}$ ^

The number of solutions of the equation $\cos 2 \theta \cos \frac{\theta}{2}+\cos \frac{5 \theta}{2}=2 \cos…

Solution