algebra precalculus - If $tan (pi cos theta) =cot (pi sin theta) $ then find the value of $cosleft (theta -frac{pi}{4}right)$

If $\tan (\pi \cos \theta) =\cot (\pi \sin \theta) $ then find the value of $\cos \left(\theta -\frac{\pi}{4}\right).$

I could not get any idea to solve. However I tried by using $\theta =0^\circ $. But could not get the answer.

Answer

Hint -

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

$\tan(\pi \cos \theta) =\tan (\frac{\pi}2 - \pi \sin \theta)$

$\pi \cos \theta =\frac{\pi}2 - \pi \sin \theta$

$\cos \theta = \frac 12 - \sin \theta$

$\sin \theta + \cos \theta = \frac 12$

Now we have,

$\cos\left(\theta-\frac{\pi}{4}\right)$

$= \cos\theta \cos \frac{\pi}4 +\sin\theta \sin \frac{\pi}4$

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

$= \frac1{\sqrt 2} \left( \cos\theta +\sin\theta \right)$

$ = \frac 1{\sqrt2} \left(\frac 12\right)$

$= \frac 1{2\sqrt2}$