Question

If $\tan \left(\pi cos\theta \right)=\cot \left(\pi sin\theta \right)$ then $19208{\cos }^2(\theta -\pi /4)$ is equal to

Answer 1

$\tan \left(\pi \cos \theta \right)=\tan (\frac{\pi }{2}-\pi \sin \theta )$
$\Rightarrow \pi \cos \theta =n\pi +\frac{\pi }{2}-\pi \sin \theta (n\in I)$
$\Rightarrow \pi (\sin \theta +\cos \theta )=(2n+1)\frac{\pi }{2}$
$\Rightarrow \sin \theta +\cos \theta =\frac{2n+1}{2}$
$\Rightarrow \cos (\frac{\pi }{4}-\theta )=\frac{2n+1}{2\sqrt{2}}$
Since $-1\le \cos (\frac{\pi }{4}-\theta )\le 1$
$\Rightarrow -1\le \frac{2n+1}{2\sqrt{2}}\le 1$
$\Rightarrow n=0,-1$ as $n$ is an integer
$\Rightarrow \cos (\frac{\pi }{4}-\theta )=\pm \left(\frac{1}{2\sqrt{2}}\right)$
$\Rightarrow 8{\cos }^2(\pi /4-\theta )=1$
$\Rightarrow 19208{\cos }^2(\pi /4-\theta )=2401$