Prove

Let $1+\cos x=t$

$\therefore-\sin x d x=d t$

$\Rightarrow \int \frac{\sin x}{1+\cos x} d x=\int-\frac{d t}{t}$

$=-\log |t|+\mathrm{C}$

$=-\log |1+\cos x|+\mathrm{C}$