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If $\sin (\pi \cos x)=\cos (\pi \sin x)$, then what is one of the values
of $\sin 2 \mathrm{x}$ ?
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of $\sin 2 \mathrm{x}$ ?
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Verified Answer
The correct answer is:
$-\frac{3}{4}$
Given that : $\sin (\pi \cos x)=\cos (\pi \sin x)$
$3 u, \cos \left(\frac{\pi}{2}-\pi \cos x\right)=\cos (\pi \sin x)$
$\Rightarrow \frac{\pi}{2}-\pi \cos x=\pi \sin x$
$\Rightarrow \sin x+\cos x=\frac{1}{2}$
Squaring both sides, we get
$\sin ^{2} x+\cos ^{2} x+2 \sin x \cos x=\frac{1}{4}$
$\Rightarrow \sin 2 x=\frac{1}{4}-1=-\frac{3}{4}$
$3 u, \cos \left(\frac{\pi}{2}-\pi \cos x\right)=\cos (\pi \sin x)$
$\Rightarrow \frac{\pi}{2}-\pi \cos x=\pi \sin x$
$\Rightarrow \sin x+\cos x=\frac{1}{2}$
Squaring both sides, we get
$\sin ^{2} x+\cos ^{2} x+2 \sin x \cos x=\frac{1}{4}$
$\Rightarrow \sin 2 x=\frac{1}{4}-1=-\frac{3}{4}$
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