Prove that `f(theta)=(4sintheta)/(2+costheta)-theta` is an increasing function of `theta` in `[0,pi/2]` .

Explanation

given:` y=frac(4sintheta)(2+cos theta)`

The derivative of y is `(8costheta+4)/(2+costheta)^2`

For y to be the increasing function, the derivative of y should be positive.

`(8costheta+4)/(2+costheta)^2>=0`,

`impliescostheta>=-1/2`

For `0<=theta<=pi/2`, `costheta` is positive . Therefore y is increasing function for `0<=theta<=pi/2`