Find the derivative of $\arctan \left( \cos x\over1+\sin x \right)$. > Find the derivative with respect to $x$ of $$\arctan \left( \cos x\over1+\sin x \right).$$ I tried solving this problem by using trigonometric functions of submultiple of numbers (formulas of $\sin x$ and $\cos x$) but they didn't help.

**Hint:**

$$\frac{\cos x}{1+\sin x} = \frac{\sin(\frac\pi2 -\frac x2 )}{1+\cos(\frac\pi2-\frac x2) } = \frac{2\sin(\frac\pi4-\frac x2)\cos(\frac\pi4-\frac x2)}{2\cos^2(\frac\pi4-\frac x2)} = \tan(\frac\pi4-\frac x2).$$