Expanding the expression $\tan x/2+\cot x/2$

Question

I'm solving a trigonometric equation and i need to expand

$$\tan \frac x2+\cot \frac x2$$

At the end in the book I see the result they get:

$$\tan \frac x2+\cot \frac x2=\frac 2{\sin x}$$

Can someone help me on that?

Answer 1

$$\tan \frac x2+\cot\frac x2=\frac{\sin \frac x2}{\cos \frac x2}+\frac{\cos \frac x2}{\sin \frac x2}=\frac{\sin^2 \frac x2+\cos^2\frac x2}{\cos \frac x2 \sin \frac x2}=\frac 2{\sin x}$$ because $\sin x=2\sin \frac x2 \cos \frac x2$ and $\sin^2 \frac x2+\cos^2 \frac x2=1$