Question

If $\sin \left(\pi \cos \theta \right)=\cos \left(\pi \sin \theta \right),then$\sin 2\theta) may take value

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

Answer 1

Answer: (B)

$\frac{-3}{4}$

$\sin \left(\pi \cos \theta \right)=\cos \left(\pi \sin \theta \right)$
$\cos (\frac{\pi }{2}-\pi \cos \theta )=\cos \left(\pi \sin \theta \right)$
$\frac{\pi }{2}-\pi \cos \theta =\pi \sin \theta$
$\frac{\pi }{2}=\pi (\cos \theta +\sin \theta )$
$\sin \theta +\cos \theta =\frac{1}{2}$ square both sides
${\sin }^2\theta +{\cos }^2\theta +2\sin \theta \cos \theta =\frac{1}{4}$
$1+\sin 2\theta =\frac{1}{4}$
$\sin 2\theta =\frac{-3}{4}$