Question

If $\tan \left(\pi \cos x\right)=\cot \left(\pi \sin x\right)$, then the absolute value of $4\sqrt{2}\cos (x-\frac{\pi }{4})$ is

Answer 1

$\tan \left(\pi \cos x\right)=\cot \left(\pi \sin x\right)\Rightarrow \tan \left(\pi \cos x\right)=\tan (\pi /2-\pi \sin x)\Rightarrow \pi \cos x=n\pi +\pi /2-\pi \sin x\Rightarrow \cos x+\sin x=\frac{2n+1}{2},n\in Z\Rightarrow \frac{1}{\sqrt{2}}\cos x+\frac{1}{\sqrt{2}}\sin x=\frac{2n+1}{2\sqrt{2}}\Rightarrow \cos (x-\frac{\pi }{4})=\pm \frac{1}{2\sqrt{2}}(\because \cos x\in [-1,1\left]\right)$
So, absolute value of $4\sqrt{2}\cos (x-\frac{\pi }{4})=2$