Trignometric identities - Prove that the following is an identity.

Hello, I am having troubles with a question involving trigonometric identities. I have always had trouble with these types of problems, so I need as much help as possible.

To the problem:

"Prove that the following is an identity: $\displaystyle \frac{cos2\theta}{1 + sin2\theta} = \frac{cot \theta - 1}{cot \theta +1} $"

Here is my attempt:

R.H.S

$\displaystyle \frac{\frac{1}{tan\theta} - 1}{\frac{1}{tan\theta} + 1} $

$\displaystyle \frac{\frac{cos\theta}{sin\theta} -1}{\frac{cos\theta}{sin\theta} +1} $

I don't know what to do from here. On the left hand start I can't see where to even start. Any help at all is very much appreciated. I really do not understand these problems.

Thank you.