Question

Solve $\cos \theta +\cos 7\theta +\cos 3\theta +\cos 5\theta =0$.

Answer 1

Given, $\cos \theta +cos7\theta +\cos 3\theta +\cos 5\theta =0$

$\Rightarrow 2\cos \left(\frac{\theta +7\theta }{2}\right)\cos \left(\frac{\theta -7\theta }{2}\right)+2\cos \left(\frac{3\theta +5\theta }{2}\right)\cos \left(\frac{3\theta -5\theta }{2}\right)=0$

$\Rightarrow 2\cos 4\theta cos3\theta +2\cos 4\theta \cos \theta =0$

$\Rightarrow 2\cos 4\theta \left(\cos 3\theta +\cos \theta \right)=0$

$\Rightarrow 2\cos 4\theta \left(2\cos \left(\frac{3\theta +\theta }{2}\right)\cos \left(\frac{3\theta -\theta }{2}\right)\right)=0$

$\Rightarrow 4\cos 4\theta \cos 2\theta \cos \theta =0$

$\Rightarrow \cos \theta =0$ or $\cos 2\theta =0$ or $\cos 4\theta =0$

$\Rightarrow \theta =(2n+1)\frac{\pi }{2}$ or $2\theta =\frac{(2n+1)\pi }{2}$ or $4\theta =\frac{(2n+1)\pi }{2}$

$\Rightarrow \theta =(2n+1)\frac{\pi }{2}or\theta =(2n+1)\frac{\pi }{4}or\theta =(2n+1)\frac{\pi }{8}$