Read the given information carefully and answer the given questions.

It is given that $$\cos (\theta - \alpha ) = a$$, $$\cos (\theta - \beta ) = b$$.

What is $${\sin ^2}(\alpha - \beta ) + 2ab\cos (\alpha - \beta )$$ equal to?

If $$\sin \alpha + \cos \alpha = p$$, then what is $${\cos ^2}(2\alpha )$$ equal to?

If $$\tan \theta = {1 \over 2}$$ and $$\tan \phi = {1 \over 3}$$, then what is the value of $$(\theta + \phi )$$ ?

If $$\cos A = {3 \over 4}$$, then what is the value of $$\sin \left( {{A \over 2}} \right)\sin \left( {{{3A} \over 2}} \right)$$ ?