Question

State whether following statement is true or false.

If $\tan \left(\pi \cos \theta \right)=\cot \left(\pi \sin \theta \right)$, then $\cos (\theta -\frac{\pi }{4})=\pm \frac{1}{2\sqrt{\left(2\right)}}$.

• A. True
• B. False

Answer 1

The correct option is A True

Given

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

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

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

$⟹\cos \theta =\frac{1}{2}-\sin \theta$

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

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

$⟹\cos \frac{\pi }{4}\cos \theta +\sin \frac{\pi }{4}\sin \theta =\pm \frac{1}{2\sqrt{2}}$

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

The given statement is true.