Question

$(\sin x+\cos x)\sqrt{2}=\tan x+\cot x$

Answer 1

$\begin{array}{c}\left(\sin x+\cos x\right)\sqrt{2}=\tan x+\cot x\\ \Rightarrow \left(\sin x+\cos x\right)\sqrt{2}=\frac{\sin x}{\cos x}+\frac{\cos x}{\sin x}\\ \Rightarrow \sqrt{2}\left(\sin x+\cos x\right)=\frac{{\sin }^{2}x+{\cos }^{2}x}{\sin x\cdot \cos x}\\ \Rightarrow \sqrt{2}\sin x\cdot \cos x\left(\sin x+\cos x\right)=1\\ \Rightarrow 2\sin x\cdot \cos x\left\{\sin \left(x+\frac{\pi }{4}\right)\right\}=1\\ \sin 2x\cdot \sin \left(x+\frac{\pi }{4}\right)=1\end{array}$

$x=\frac{\pi }{4}$