Aspire Faculty ID #3662 · Topic: NIMCET 2012 · Just now
NIMCET 2012

If $\sin(\pi\cos\theta)=\cos(\pi\sin\theta)$, then $\sin 2\theta$ is equal to

Solution

Given, $\sin(\pi\cos\theta)=\cos(\pi\sin\theta)$ 

Using, $\cos x=\sin\left(\frac{\pi}{2}-x\right)$ 

Therefore, $\sin(\pi\cos\theta)=\sin\left(\frac{\pi}{2}-\pi\sin\theta\right)$ 

This gives the following two possible cases. 

Case 1: $\pi\cos\theta=\frac{\pi}{2}-\pi\sin\theta$ 
$\cos\theta+\sin\theta=\frac{1}{2}$ 
Squaring both sides, $(\cos\theta+\sin\theta)^2=\frac{1}{4}$ 

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

$1+\sin 2\theta=\frac{1}{4}$ 
$\sin 2\theta=-\frac{3}{4}$ 

Case 2: 
$\pi\cos\theta=\pi-\left(\frac{\pi}{2}-\pi\sin\theta\right)$ 
$\cos\theta-\sin\theta=\frac{1}{2}$ 
Squaring both sides, 
$(\cos\theta-\sin\theta)^2=\frac{1}{4}$ 
$1-\sin 2\theta=\frac{1}{4}$ 
$\sin 2\theta=\frac{3}{4}$ 
Hence, $\sin 2\theta=\pm\frac{3}{4}$ 
 Therefore, the correct answer is: $\boxed{\text{(a) }\pm\frac{3}{4}}$
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