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

In terms of $t=\tan\tfrac{x}{2}$, the LHS is $t\cdot\frac{1+t^2}{1-t^2}$, while the RHS is $\frac{2t}{1-t^2}-t=t\cdot\frac{2-(1-t^2)}{1-t^2}=t\cdot\frac{1+t^2}{1-t^2}$.
