Упростите выражение

Решите задачу:

$\dfrac{1}{\mathrm{tg}(x)}+\dfrac{\sin(x)}{1+\cos(x)}=\dfrac{\cos(x)}{\sin(x)}+\dfrac{\sin(x)}{1+\cos(x)}=\dfrac{\cos(x)\left(1+\cos(x)\right)+\sin^2(x)}{\sin(x)\left(1+\cos(x)\right)}=\medskip\\=\dfrac{\cos(x)+\cos^2(x)+\sin^2(x)}{\sin(x)\left(1+\cos(x)\right)}=\dfrac{\left(\cos(x)+1\right)}{\sin(x)\left(1+\cos(x)\right)}=\dfrac{1}{\sin(x)}=\mathrm{cosec}(x)$