Question

The smallest positive integral value of p for which the equation $\cos \left(p\sin x\right)=\sin \left(p\cos x\right)$ in x has a solution in $[0,2\pi ]$ is

• A. 2
• B. 1
• C. 3
• D. none of these

Answer 1

The correct option is A 2

$\cos \left(p\sin x\right)=\sin \left(p\cos x\right)$

$\Rightarrow \cos \left(p\sin x\right)=\cos (\frac{\pi }{2}-p\cos x)$

$\Rightarrow p\sin x=\frac{\pi }{2}-p\cos x\Rightarrow \cos x+\sin x=\frac{\pi }{2p}$

condition for this equation to have solution is,
$\frac{\pi }{2p}\le \sqrt{2}\Rightarrow p\ge \frac{\pi }{2\sqrt{2}}$

Hence smallest positive integer is $2$.