Question

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

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

Answer 1

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

$\tan \left(\pi \cos \theta \right)$

$=\cot \left(\pi \sin \theta \right)=\tan (\pm \frac{\pi }{2}-\pi \sin \theta )$

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

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