Find the number of solutions of the equation ` 1+e^( cot^(2)x)=sqrt(2|sinx|-1)+(1- cos 2x )/(1+ sin^(4)x) ` for ` x in (0,5pi)`

Explanation

`L.H.S. =1+e^(cot^(2) x) ge 2`
As `sqrt(2|sin x|-1) le 1`
and `(1- cos 2x)/(1+sin^(4) x)=(2 sin^(2) x)/(1+sin^(4) x)=2/(1/(sin^(2) x)+sin^(2) x) le 1`
`:. R.H.S.=sqrt(2|sin x|-1)+(1- cos 2x)/(1+sin^(4) x) le 2`
Equation will be satisfied if `L.H.S.=R.H.S.=2`.
This is possible when `cot^(2) x=0` and `|sin x|=1`.
`rArr x=(2n+1) pi.2, n in Z`