Question

Let $X=\left\{x\in R:\cos \left(\sin x\right)=\sin \left(\cos x\right)\right\}$ The number of elements in $X$ is

• A. 0
• B. 2
• C. 4
• D. not finite

Answer 1

Answer: (A)

We have,
$X=\left\{x\in R:\cos \left(\sin x\right)=\sin \left(\cos x\right)\right\}$
$\cos (\sin x)=\sin (\cos x)$
$\sin \left(\frac{\pi }{2}\pm \sin x\right)=\sin \left(\cos x\right)$
$\Rightarrow \cos x=\frac{\pi }{2}\pm \sin x$
$\Rightarrow \cos x=n\pi +{\left(-1\right)}^n\left(\frac{\pi }{2}+\sin x\right),n\in I$
$\Rightarrow \cos x\pm \sin x=n\pi +{\left(-1\right)}^n\frac{\pi }{2},n\in I$
As $LHS\in \left[-\sqrt{2},\sqrt{2}\right]$, and it does not satisfy the RHS.
$\therefore$ No solution exists