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  4. If tan (π cos θ)= cot (π sin θ), then a value of cos (θ-(π/4)) among the following is

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Q. If $\tan (\pi \cos \theta)=\cot (\pi \sin \theta)$, then a value of $\cos \left(\theta-\frac{\pi}{4}\right)$ among the following is

EAMCETEAMCET 2013

Solution:

Given, $\tan (\pi \cos \theta)=\cot (\pi \sin \theta)$
$\Rightarrow \tan (\pi \cos \theta)=\tan \left\{\frac{\pi}{2}-\pi \sin \theta\right\}$
$\Rightarrow \pi \cos \theta=\frac{\pi}{2}-\pi \sin \theta$
$\Rightarrow \sin \theta+\cos \theta=\frac{1}{2}$
$\Rightarrow \frac{1}{\sqrt{2}} \sin \theta+\frac{1}{\sqrt{2}} \cos \theta=\frac{1}{2 \sqrt{2}}$
$\Rightarrow \cos \theta \cdot \cos \frac{\pi}{4}+\sin \theta \cdot \sin \frac{\pi}{4}=\frac{1}{2 \sqrt{2}}$
$\Rightarrow \cos \left(\theta-\frac{\pi}{4}\right)=\frac{1}{2 \sqrt{2}}$