Solve `8sinx=(sqrt(3))/(cosx)+1/(sinx)`

Explanation

`8 sin x=sqrt(3)/(cos x)+1/(sin x)`
or `8 sin^(2) x cos x=sqrt(3) sin x + cos x`
or `4(2 sin x cos x) sin x =sqrt(3)sin x+ cos x`
or `2(2 sin 2x sin x)=sqrt(3) sin x+ cos x`
or `2 cos x-2 cos 3x =sqrt(3) sin x+ cos x`
or `cos x-sqrt(3) sin x =2 cos 3x`
or `cos 3x=cos(x+pi/3)`
`rArr 3x=2n pi pm (x+pi/3), n in Z`
By taking positive sign, `x= npi+pi/6, n in Z`
By taking negative sign, `x=(npi)/2-pi/12, n in Z`