If $$\sin x + \sin y = a$$ and $$\cos x + \cos y = b$$, then $${\tan ^2}\left( {{{x + y} \over 2}} \right) + {\tan ^2}\left( {{{x - y} \over 2}} \right)$$ is equal to

If $$p = \tan \left( { - {{11\pi } \over 6}} \right)$$, $$q = \tan \left( {{{21\pi } \over 4}} \right)$$ and $$r = \cot \left( {{{283\pi } \over 6}} \right)$$, then which of the following is/are correct?

1. The value of p $$\times$$ r is 2.

2. p, q and r are in GP.

Select the correct answer using the code given below.

$$\sin A + 2\sin 2A + \sin 3A$$ is equal to which of the following?

I. $$4\sin 2A{\cos ^2}\left( {{A \over 2}} \right)$$

II. $$2\sin 2A{\left( {\sin {A \over 2} + \cos {A \over 2}} \right)^2}$$

III. $$8\sin A\cos A{\cos ^2}\left( {{A \over 2}} \right)$$

Select the correct answer using the code given below.

If $$x = \sin 70^\circ \sin 50^\circ $$ and $$y = \cos 60^\circ \cos 80^\circ $$, then what is xy equal to?