Question

Assertion :Statement 1: The equation $\sin \left(\cos x\right)=\cos \left(\sin x\right)$ has no real solution Reason: Statement 2: $\sin x\pm \cos xϵ[-\sqrt{2},\sqrt{2}]$

• A. if both the statements are TRUE and STATEMENT 2 is the correct explanation of STATEMENT 1
• B. if both the statements are TRUE and STATEMENT 2 is NOT the correct explanation of STATEMENT 1
• C. if STATEMENT 1 is TRUE and STATEMENT 2 is FALSE
• D. if STATEMENT 1 is FALSE and STATEMENT 2 is TRUE

Answer 1

The correct option is A if both the statements are TRUE and STATEMENT 2 is the correct explanation of STATEMENT 1

statement 1.
$sin\left(cosx\right)=cos\left(sinx\right)$

$\Rightarrow cos\left(sinx\right)=cos(2n\pi +\frac{\pi }{2}-cosx)$, where n is any integer

Therefore general solution is,

$sinx=2m\pi \pm (2n\pi +\frac{\pi }{2}-cosx)$, where m is any integer

$\Rightarrow sinx\pm cosx=(m\pm n)2\pi \pm \frac{\pi }{2}$

we can see that for any combination of integer pair $(m,n)$

$\mid sinx\pm cosx\mid >\sqrt{2}$

but we now, $\mid sinx\pm cosx\mid \le \sqrt{2}$

.thus there won't exist any solution.

Hence statement 1 is correct.

we obviously know that $sinx\pm cosxϵ[-\sqrt{2},\sqrt{2}]$

thus statement 2 is also correct and it is well explaining statement 1.

Hence option 'A' is correct choice.