Question

The number of solutions to $\cos (\sin x)=\sin (\cos x),x\in [0,20\pi ]$, is

• A. 5
• B. 10
• C. 15
• D. None of these

Answer 1

Answer: (D)

$\cos (\sin x)=\sin (\cos x)$
$\Rightarrow \cos (\sin x)=\cos (\pi /2-\cos x)$
$\Rightarrow \sin x=2n\pi \pm (\pi /2-\cos x)$
Taking positive sign, $\sin x=2n\pi +\pi /2-\cos x$
or $(\cos x+\sin x)=\frac{1}{2}(4n+1)\pi$
Now $L.H.S.\in [-\sqrt{2},\sqrt{2}]$,
hence $-\sqrt{2}\le \frac{1}{2}(4n+1)\pi \sqrt{2}$
$\Rightarrow \frac{-2\sqrt{2}-\pi }{4\pi }\le n\le \frac{2\sqrt{2}-\pi }{4\pi }$,
which is not satisfied by any integer $n$.
Similarly, taking negative sign we have $\sin x-\cos x=(4n-1)\pi /2$,
which is also not satisfied for any integer $n$.
Hence, there is no solution.