Question

If $\sin (\pi \cos \theta )=\cos (\pi \sin \theta ),$ then $\sin 2\theta$ equals

• A. $\pm \frac{3}{4}$
• B. $\pm \sqrt{2}$
• C. $\pm \frac{1}{\sqrt{3}}$
• D. $\pm \frac{1}{2}$

Answer 1

Answer: (A)

$\sin \left(\pi \cos \theta \right)=\cos \left(\pi \sin \theta \right)$
$\Rightarrow \cos \left(\frac{\pi }{2}-\pi \cos \theta \right)=\cos \left(\pi \sin \theta \right)$
$\Rightarrow \frac{\pi }{2}-\pi \cos \theta -\pm \pi \sin \theta$
$\Rightarrow \frac{\pi }{2}=\pi \cos \theta \pm \pi \sin \theta$
$\frac{1}{2}=\cos \theta \pm \sin \theta$
Squaring both sides, we get
$\frac{1}{4}={\cos }^2\theta +{\sin }^2\theta \pm 2\sin \theta \cos \theta$
$\Rightarrow \frac{1}{4}=1\pm 2\sin \theta \cos \theta$
$\Rightarrow \frac{1}{4}-1\pm \sin 2\theta$
$\Rightarrow -\frac{3}{4}=\pm \sin 2\theta \Rightarrow \sin 2\theta \pm \frac{3}{4}$