Question

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

Answer 1

$y^{\prime }=\frac{8c+4}{(c+2{)}^2}-1=\frac{\cos \theta (4-\cos \theta )}{(\cos \theta +2{)}^2}$

for increasing it should be $y^{\prime }>0$ we can say that $(c+2{)}^2;4-c$ are always $>0$ and the other will be positive in first quadrant. hence it increases in $[0,\frac{\pi }{2}]$