Mathematics - Trigonometric Equation Question with Solution | TestHub

$\begin{array}{rl}& \cos 2\theta \cos \frac{\theta }{2}+\cos \frac{5\theta }{2}=2{\cos }^3\frac{5\theta }{2}\\ & \frac{1}{2}\left(2\cos 2\theta \cos \frac{\theta }{2}\right)+\cos \frac{50}{2}\\ & =\frac{1}{2}\left(\cos \frac{15\theta }{2}+3\cos \frac{5\theta }{2}\right)\end{array}$
or solving
$\begin{array}{rl}& \cos \frac{3\theta }{2}=\cos \frac{15\theta }{2}\\ & \cos \frac{15\theta }{2}-\cos \frac{3\theta }{2}=0\\ & 2\sin 30\sin \frac{9\theta }{2}=0\end{array}$
$3\theta =\mathrm{n}\pi \text{ or }\frac{9\theta }{2}=\mathrm{m}\pi$
$\begin{array}{rl}& \theta =\frac{\mathrm{n}\pi }{3}\theta =\frac{2\mathrm{m}\pi }{9}\\ & \theta =\left\{-\frac{\pi }{2},\frac{\pi }{3},0\right\}\\ & \theta =\left\{-\frac{4\pi }{9},\frac{-2\pi }{9},\frac{4\pi }{9},\frac{2\pi }{9}\right\}\end{array}$