Question

Assertion :The number of real solution of the equation $\sin \left(\cos x\right)=\cos \left(\sin x\right)$ is zero. Reason: $\sin x>0$,then $2n\pi <x<(2n+1)\pi ,nϵI$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion false but Reason is true

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

As $\sin \left(\cos x\right)=\cos \left(\sin x\right)$
$\Rightarrow \cos \left(\sin x\right)=\cos (\frac{\pi }{2}-\cos x)$
$\Rightarrow \sin x=2n\pi \pm \frac{\pi }{2}-\cos x,n\in I$

$\Rightarrow \cos x+\sin x=(2n\pm \frac{1}{2})\pi$

On squaring, we get
$\Rightarrow 1+\sin 2x={(2n\pm \frac{1}{2})}^2{\pi }^2$

$\Rightarrow \left|\sin 2x\right|={(2n\pm \frac{1}{2})}^2{\pi }^2-1$

$\Rightarrow \left|\sin 2x\right|>1\forall n\in I$
which is not possible

$\therefore$ Given equation has no real solution
Again $\sin x$ is positive so $x$ must lie in first or second quadrant

$2n\pi <x<(2n+1)\pi ,n\in I$