What is the simplified value of 1 + cot A cot $$(\frac{A}{2})$$?

Now We have :
$$1+\cot A\ \cot\ \frac{A}{2}$$
Now we get 1+$$\frac{\cos A}{\sin A}\times\ \frac{\cos\ \frac{A}{2}}{\sin\ \frac{A}{2}}=\ \frac{\cos A}{2\sin\ \frac{A}{2}\cos\ \frac{A}{2}}\times\ \frac{\cos\ \frac{A}{2}}{\sin\ \frac{A}{2}}$$
we get $$1+\frac{\cos A}{2\sin^2\ \frac{A}{2}}$$
Now cos A = $$2\sin^2\ \frac{A}{2}-1$$
substituting we get $$\frac{1}{2\sin^2\ \frac{A}{2}}=\frac{1}{2}\operatorname{cosec}^2\ \frac{A}{2}$$