What is $\cot(\dfrac{A}{2})-\tan(\dfrac{A}{2})$ equal to ?

$\cot \left(\dfrac{A}{2}\right)-\tan\left(\dfrac{A}{2}\right)$

$\Rightarrow$ $\dfrac{\cos \dfrac{A}{2}}{\sin\dfrac{A}{2}}-\dfrac{\sin \dfrac{A}{2}}{\cos \dfrac{A}{2}}$

$\Rightarrow$ $\dfrac{\cos^2\dfrac{A}{2}-\sin^2\dfrac{A}{2}}{\sin\dfrac{A}{2}.\cos\dfrac{A}{2}}$

$\Rightarrow$ $\dfrac{\cos 2\left(\dfrac{A}{2}\right)}{\sin\dfrac{A}{2}.\cos\dfrac{A}{2}}$

$\Rightarrow$ $\dfrac{\cos A}{\sin\dfrac{A}{2}.\cos\dfrac{A}{2}}$

$\Rightarrow$ $\dfrac{2\cos A}{2\sin\dfrac{A}{2}.\cos\dfrac{A}{2}}$

$\Rightarrow$ $\dfrac{2\cos A}{\sin 2\left(\dfrac{A}{2}\right)}$

$\Rightarrow$ $\dfrac{2\cos A}{\sin A}$

$\Rightarrow$ $2\cot A$