Question

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

Answer 1

$\tan \left(\pi \cos \theta \right)=\cot \left(\pi \sin \theta \right)$

$\frac{\sin \left(\pi \cos \theta \right)}{\cos \left(\pi \cos \theta \right)}=\frac{\cos \left(\pi \sin \theta \right)}{\sin \left(\pi \sin \theta \right)}$

$\sin \left(\pi \cos \theta \right)\sin \left(\pi \sin \theta \right)=\cos \left(\pi \cos \theta \right)\cos \left(\pi \sin \theta \right)$

$\cos \left(\pi \cos \theta \right)\cos \left(\pi \sin \theta \right)-\sin \left(\pi \cos \theta \right)\sin \left(\pi \sin \theta \right)=0$

$\cos (\pi \cos \theta +\pi \sin \theta )=0$

$\pi \cos \theta +\pi \sin \theta =\pm \frac{\pi }{2}$

$\cos (\pm \frac{\pi }{2})=0$

$\cos \theta +\sin \theta =\pm \frac{1}{2}$

Multiply both sides by $\frac{1}{\sqrt{2}}$

$\frac{1}{\sqrt{2}}\cos \theta +\frac{1}{\sqrt{2}}\sin \theta =\pm \frac{1}{2\sqrt{2}}$

$\cos \frac{\pi }{4}\cos \theta +\sin \frac{\pi }{4}\sin \theta =\pm \frac{1}{2\sqrt{2}}$

$\cos (\theta -\pi /4)=\pm \frac{1}{2\sqrt{2}}$

Hence Proved.