Question

If $\sin \left(\pi \cos \theta \right)=\cos \left(\pi \sin \theta \right)$, then the value of $\cos (\theta +\frac{\pi }{4})$ is

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

Answer 1

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

We have $\sin \left(\pi \cos \theta \right)=\cos \left(\pi \sin \theta \right)=\sin (\frac{\pi }{2}+\pi \sin \theta )$

$\Rightarrow \pi \cos \theta =\frac{\pi }{2}+\pi \sin \theta \Rightarrow \pi (\cos \theta -\sin \theta )=\frac{\pi }{2}$

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

$\Rightarrow \cos \theta \cos \cdot \frac{\pi }{4}-\sin \theta \cdot \sin \frac{\pi }{4}=\frac{1}{2}\cdot \frac{1}{\sqrt{2}}$

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

$\Rightarrow \cos (\theta +\frac{\pi }{4})=\frac{1}{2\sqrt{2}}$