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 $$\left[ {0,2\pi } \right]$$  is

Solution :
$$\eqalign{ & {\text{Here,}}\,\,p\,\sin x = \frac{\pi }{2} \pm p\cos x\,\,\,{\text{or,}}\,\,\sin x \mp \,\cos x = \frac{\pi }{{2\,p}} \cr & {\text{or,}}\,\,{\text{sin}}\left( {x \mp \frac{\pi }{4}} \right) = \frac{\pi }{{2\sqrt {2\,p} }} \cr & \Rightarrow \,\left| {\frac{\pi }{{2\sqrt {2\,p} }}} \right| \leqslant 1 \cr} $$
∴ for positive $$p,p \geqslant \frac{\pi }{{2\sqrt 2 }}.\,{\text{But}}\,{\text{1}}\,{\text{ < }}\frac{\pi }{{2\sqrt 2 }} < 2.$$