Question

Solve :- $cos\theta +sin\theta =cos2\theta +sin2\theta$

Answer 1

We have,

$\cos \theta +\sin \theta =\cos 2\theta +\sin 2\theta$

$\Rightarrow \cos \theta -\cos 2\theta =\sin 2\theta -\sin \theta$

$\Rightarrow 2\sin \frac{\theta +2\theta }{2}\sin \frac{2\theta -\theta }{2}=2\cos \frac{2\theta +\theta }{2}\sin \frac{2\theta -\theta }{2}$

$\Rightarrow \sin \frac{3\theta }{2}=\cos \frac{3\theta }{2}$

$\Rightarrow \frac{\sin \frac{3\theta }{2}}{\cos \frac{3\theta }{2}}=1$

$\Rightarrow \tan \frac{3\theta }{2}=1$

$\Rightarrow \tan \frac{3\theta }{2}=\tan \frac{\pi }{4}$

$\Rightarrow \frac{3\theta }{2}=n\pi +\frac{\pi }{4}$

$\Rightarrow 3\theta =2n\pi +\frac{\pi }{2}$

$\Rightarrow \theta =\frac{2}{3}n\pi +\frac{\pi }{6}$

Hence, this is the answer.