Question

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

• A. $\cos \frac{\pi }{4}$
• B. $\frac{1}{2}\cos \frac{\pi }{4}$
• C. $\cos \frac{\pi }{8}$
• D. None of these

Answer 1

The correct option is B $\frac{1}{2}\cos \frac{\pi }{4}$

We have, $\sin \left(\pi \cos \theta \right)=\cos \left(\pi \sin \theta \right)$
$\Rightarrow \sin \left(\pi \cos \theta \right)=\sin (\frac{\pi }{2}\pm \pi \sin \theta )$
$\Rightarrow \pi \cos \theta =\frac{\pi }{2}\pm \pi \sin \theta$
$\Rightarrow \cos \theta \pm \sin \theta =\frac{1}{2}$
$\Rightarrow \frac{1}{\sqrt{2}}\cos \theta \pm \frac{1}{\sqrt{2}}\sin \theta =\frac{1}{2\sqrt{2}}$
$\Rightarrow \cos (\theta \pm \frac{\pi }{4})=\frac{1}{2\sqrt{2}}$
$\Rightarrow \cos (\theta \pm \frac{\pi }{4})=\frac{1}{2}\cos \frac{\pi }{4}$.