Q. If $\tan{(\pi\cos{\theta) = \cot{(\pi\sin{\theta)}}}}$, then the value of $\cos\left( \theta - \frac{\pi}{4} \right)$ is equal to
A) $\frac{1}{2\sqrt{2}}$
B) $\frac{1}{\sqrt{2}}$
C) $\frac{1}{3\sqrt{2}}$
D) $\frac{1}{4\sqrt{2}}$
Explanation
(a)
We have, $\tan{(\pi\cos{\theta) = \tan\left( \frac{\pi}{2} - \pi\sin\theta \right)}}$
$\therefore\ \sin{\theta + \cos{\theta = \frac{1}{2}}}$
$\Rightarrow \ \frac{1}{\sqrt{2}}\cos\theta + \frac{1}{\sqrt{2}}\sin{\theta = \frac{1}{2\sqrt{2}}}$
$\Rightarrow \ \cos{\left( \ \theta - \frac{\pi}{4} \right) =}\frac{1}{2\sqrt{2}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.