trigonometry - Circle with cyclic quadrilateral angles with tangent

I am trying to answer the following questions. As you can see, I have done the first part but cannot make headway with the second part.

PTQ is the tangent to a circle at the point T. The points A and B on the circumference of the circle are such that TA and TB make acute angles $\alpha$ and $\beta$ with TP and TQ respectively. If AB meets the diameter through T at N, prove that TN = $\frac{a \sin\alpha\sin\beta}{\cos(\alpha - \beta)}$, where a is the length of the diameter.
If the points C and D on the circumference of the circle are such that TC and TD make acute angles $\gamma$ and $\delta$ with TP and TQ respectively and CD meets the diameter through T at the same point N, prove that $\tan \alpha\tan\beta = \tan\gamma\tan \delta$

Answer

Here is my final working for the problem