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

Explanation

To prove that the function \( y = \frac{4 \sin \theta}{2 + \cos \theta} - \theta \) is an increasing function of \( \theta \) in the interval \( [0, \frac{\pi}{2}] \), we need to show that the derivative \( y' \) is greater than or equal to 0 for all \( \theta \) in this interval.

### Step 1: Differentiate the function
We start by differentiating \( y \):
\[ y' = \frac{d}{d\theta} \left( \frac{4 \sin \theta}{2 + \cos \theta} \right) - \frac{d}{d\theta}(\theta) \]
Using the quotient rule for differentiation, where \( u = 4 \sin \theta \) and \( v = 2 + \cos \theta \):
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