Question

If $\sin \left(\pi \cos x\right)=\cos \left(\pi \sin x\right)$, then $\sin 2x=$

• A. $-\frac{3}{4}$
• B. $-\frac{4}{3}$
• C. $\frac{1}{3}$
• D. none of these

Answer 1

Answer: (A)

$-\frac{3}{4}$

Given $\sin \left(\pi \cos x\right)=\cos \left(\pi \sin x\right)$

$\sin \left(\pi cosx\right)=\sin (\frac{\pi }{2}-\pi \sin x)$

$\pi \cos x=\frac{\pi }{2}-\pi \sin x$

$\cos x+\sin x=\frac{1}{2}$

Squaring on both sides

${\sin }^{2}x+{\cos }^{2}x+2\sin x\cos x=\frac{1}{4}$

$1+2\sin x\cos x=\frac{1}{4}$

$\sin 2x=-\frac{3}{4}$