Prove: $\tan{\frac{x}{2}}\sec{x}= \tan{x} – \tan{\frac{x}{2}}$

Question

I was solving a question which required the above identity to proceed but I never found its proof anywhere. I tried to prove it but got stuck after a while.

I reached till here:

To Prove: $$\tan{\frac{x}{2}}\sec{x}= \tan{x} – \tan{\frac{x}{2}}$$

But I don't know what to do next.
Any help is appreciated
Thanks

Answer 1

$$\tan \dfrac x2\sec x=\dfrac{\sin\dfrac x2}{\cos\dfrac x2\cdot\cos x}$$

Use $\sin\dfrac x2=\sin\left(x-\dfrac x2\right)=?$