Question

$\cos \theta +\sin \theta =\cos 2\theta +\sin 2\theta$ Find the value of $\theta$.

Answer 1

$(\cos \theta -\cos 2\theta )-(\sin 2\theta -\sin \theta )=0$
$2\sin \left(3\theta /2\right)\sin \left(\theta /2\right)-2\cos \left(3\theta /2\right)\sin \left(\theta /2\right)=0$
or $\sin \left(\theta /2\right)\left\{\sin \right(3\theta /2)-\cos (3\theta /2\left)\right\}=0$
$\sin \left(\theta /2\right)=0$ $\therefore \theta /2=n\pi$ or $\theta =2n\pi$
$\sin \left(3\theta /2\right)-\cos \left(3\theta /2\right)=0$
$\therefore \tan \left(3\theta /2\right)=1=\tan \left(\pi /4\right)$
$\therefore \frac{3\theta }{2}=n\pi +\frac{\pi }{4}$ or $\theta =\frac{2n\pi }{3}+\frac{\pi }{6}$.