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Prove that y=(4sintheta)/((2+costheta))-...

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

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`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]`.
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