Question

If tan($\pi$ cos x) = cot ($\pi$ sin x), prove that $cos(x-\frac{\pi }{4})=\pm \frac{1}{2\sqrt{2}}$

Answer 1

$\tan \left(\pi \cos x\right)=\cot \left(\pi sinx\right)\tan \left(\pi \cos x\right)=\tan (\frac{\pi }{2}-\pi \sin x)\cos x=\frac{1}{2}-\sin x\cos x+\sin x=\frac{1}{2}\frac{1}{\sqrt{2}}\cos x+\frac{1}{\sqrt{2}}\sin x=\pm \frac{1}{2\sqrt{2}}\cos \frac{\pi }{4}\cos x+\sin \frac{\pi }{4}\sin x=\pm \frac{1}{2\sqrt{2}}\cos (x-\frac{\pi }{4})=\pm \frac{1}{2\sqrt{2}}$