Prove that ` y = ( 4 sin theta)/(2 + cos theta) - theta` is an increasing function of `theta in [0,pi/2]`

Explanation

`y=(4 sin theta)/(2+ cos theta) - theta`
`(2+cos theta)* 4 cos theta`
` rArr (dy)/(d theta) = (-4 sin theta * (-sin theta))/((2 + cos theta )^(2))`
`= (8 cos theta + 4 cos^(2) theta + 4 sin^(2) theta)/((2+ cos theta)^(2))- 1`
` = (4+ 8 cos theta )/((2+ cos theta)^(2)) -1 = (4+8 cos theta- (2+cos theta)^(2))/((2+ cos theta)^(2))`
`=(4 cos theta - cos^(2) theta)/(2+ cos theta) = (cos theta(4- cos theta))/((2+ cos theta)^(2))`
Now, `(dy)/(d theta) ge 0`
`rArr (cos theta(4- cos theta))/((2+ cos theta )^(2)) ge 0`
` rArr cos theta ge 0 " "(.:' 4- cos theta gt 0 )`
` rArr theta in [0,pi/2]`
Therefore, the function is increasing in ` [0, pi/2]`.