2016/083) Find the smallest positive p such that $\cos(p\sin\, x) = \sin (p\cos\,x)$

we have
$\cos(p\sin\, x) = \sin (p\cos\,x)= \cos (\frac{\pi}{2} -  p\cos\,x)$
hence
$p\sin \,x  = \frac{\pi}{2} - p \cos \,x$ (other values shall given -ve / larger p)
hence
$p(\cos \,  x + \sin\, x) = \frac{\pi}{2}$
or $\sqrt{2}p( \sin \frac{\pi}{4} \cos \, x + \cos \frac{\pi}{4} \sin \, x) = \frac{\pi}{2}$
or $\sqrt{2}p( \sin ( x + \frac{\pi}{4}) = \frac{\pi}{2}$
the largest value of $\sin ( x + \frac{\pi}{4})$ is 1 hence smallest $p$ is $\frac{\pi}{2\sqrt{2}}$