Question

If $\tan \left(\pi \cos \theta \right)=\cot \left(\pi \sin \theta \right)$, then $\sin (\theta +\frac{\pi }{4})$ equals

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

Answer 1

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

Let us consider the problem:

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

Implies that

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

Implies that

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

Implies that

$\frac{1}{2}=\frac{1}{\sqrt{2}}[\sin \frac{\pi }{4}\cos \theta +\cos \frac{\pi }{4}\sin \theta ]$

Implies that

$\frac{1}{\sqrt{2}}=\sin \left(\frac{\pi }{4}+\theta \right)$

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

Implies that

$\theta =0$

Hence,

$\sin \left(\theta +\frac{\pi }{4}\right)=\frac{1}{\sqrt{2}}$