Question

Solve $\cos \left(\sin \theta \right)=\sin \left(\cos \theta \right)$.

Answer 1

$\cos \left(\sin \theta \right)=\cos \{\frac{\pi }{2}-\cos \theta \}$
$\therefore \sin \theta =2n$ $\pi \pm (\frac{\pi }{2}-\cos \theta )$
(i) Taking $+$ive sign, $\cos \theta +\sin \theta =(4n+1)\frac{\pi }{2}$
or $\cos (\theta -\frac{\pi }{4})=(4n+1)\frac{\pi }{2\sqrt{2}},\frac{\pi }{2\sqrt{2}}>1$
The R.H.S. is $>1$ for $n\ge 0$ and $<-1$ for $n<0$ Hence the equation does not possess any solution.
(ii)Taking $-$ive sign,
$\sin \theta -\cos \theta =(4n-1)\frac{\pi }{2}$
$\therefore \sin (\theta -\frac{\pi }{4})=(4n-1)\frac{\pi }{2\sqrt{2}}$
R.H.S. is either $>1$ for $<-1$.
Hence the equation does not possess any solution in this case also.