A problem on range of a trigonometric function: what is the range of $\frac{\sqrt{3}\sin x}{2+\cos x}$? What is the range of the function 
$$\frac{\sqrt{3}\sin x}{2+\cos x}$$ 

Let $$y = \frac{\sqrt{3}\sin x}{2+\cos x}\Rightarrow 2y+y\cos x = \sqrt{3}\sin x$$
So $$\sqrt{3}\sin x-y\cos x = 2y$$
Now Using $\bf{Cauchy\; Schwarz \; Inequality}$
$$\left[(\sqrt{3})^2+(-1)^2\right]\cdot \left[\sin^2 x+\cos^2 x\right]\geq \left[\sqrt{3}\sin x-y\cos x\right]^2$$
So $$4\geq 4y^2\Rightarrow y^2\leq 1\Rightarrow y \in \left[-1,1\right]$$
