Question

Prove that $y=\frac{4\sin \theta }{\left(2+\cos \theta \right)}-\theta$ is an increasing on $\theta$ in $\left[0,\frac{\pi }{2}\right]$

Answer 1

$\cos \theta$ is positive in $\left(0to\frac{\pi }{x}\right)$

$4-\cos \theta >0$

As $\theta -1\le \cos \theta \le 1$

Denominator is perfect square so it will be positive always.

$y^1>0$

So, $y=\frac{4\sin \theta }{2+\cos \theta }-\theta$ is increasing

In $\left(0,\frac{\pi }{2}\right)$