Question

Given $\tan \left(\pi cos\theta \right)=\cot \left(\pi sin\theta \right)$, then the value of $cos\left(\begin{array}{c}\theta -\frac{1}{4}\pi \end{array}\right)$ will be

• A. $\frac{1}{2\sqrt{2}}$
• B. $\frac{1}{\sqrt{2}}$
• C. $\frac{1}{3\sqrt{2}}$
• D. $\frac{1}{4\sqrt{2}}$

Answer 1

The correct option is A $\frac{1}{2\sqrt{2}}$

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