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