Question

Find the common roots of the equations
$\cos 2x+\sin 2x=\cot x$
and $2{\cos }^{2}x+{\cos }^{2}2x=1$

Answer 1

$\begin{array}{c}\cos 2x+\sin 2x=\cot x\\ \Rightarrow 1-2{\sin }^{2}x+2\sin x.\cos x=\frac{\cos x}{\sin x}\\ \Rightarrow \sin x-2{\sin }^{3}x+2{\sin }^{2}x\cos x=\cos x\\ \Rightarrow \sin x\left(1-2{\sin }^{2}x\right)=\cos x\left(1-2{\sin }^{2}x\right)\\ \Rightarrow 1-2{\sin }^{2}x=0\&\sin x-\cos x=0\\ \Rightarrow \sin x=\pm \frac{1}{\sqrt{2}}\&\sin x=\cos x,Sox=\frac{\pi }{4}\\ Now,\\ 2{\cos }^{2}x+{\cos }^2\left(2x\right)=1\\ \Rightarrow {\cos }^2\left(2x\right)=1-2{\cos }^{2}x=-\cos 2x\\ \Rightarrow \cos 2x=-1\&\cos 2x=0\\ \Rightarrow x=\frac{\pi }{2}andx=\frac{\pi }{4}\\ Hence,thecommonrootis\frac{\pi }{4}.\end{array}$