Question

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

• A. $\frac{1}{\sqrt{7}}$
• B. $\sqrt{7}$
• C. $\frac{2}{\sqrt{7}}$
• D. $2\sqrt{7}$

Answer 1

The correct option is A $\frac{1}{\sqrt{7}}$

$\tan \left(\pi \sin \theta \right)=\cot \left(\pi \cos \theta \right)$
$\Rightarrow \frac{\sin \left(\pi \sin \theta \right)}{\cos \left(\pi \sin \theta \right)}=\frac{\cos \left(\pi \cos \theta \right)}{\sin \left(\pi \cos \theta \right)}$
$\Rightarrow \cos (\pi \cos \theta +\pi \sin \theta )=0$
$\Rightarrow \cos \theta +\sin \theta =\pm \frac{1}{2}$
$\Rightarrow \cos (\theta -\frac{\pi }{4})=\pm \frac{1}{2\sqrt{2}}$
$\Rightarrow \cot (\theta -\frac{\pi }{4})=\pm \frac{1}{\sqrt{7}}$