MHB Show the trigonometric equation has no solutions

[anemone]

Show that the trigonometric equation $\sin (\cos a)= \cos (\sin a)$ has no solutions.

 

[kaliprasad]

Show that the trigonometric equation $\sin (\cos a)= \cos (\sin a)$ has no solutions.

Spoiler

if sin (x) = cos (y) then

$x+y = \pi/2 +2npi$ or $(2n+1)pi-\pi/2$

so lowest | cos a + sin a | = $\pi/2$

so |cos a + sin a| =$ \pi/2$ should have a solution

but |cos a + sin a | <= $\sqrt(2)$

so no solution