4. (a) Use the identities for ( \(\sin A + \sin B\) ) and ( \(\cos A + \cos B\) ) to prove that $$\frac { \sin 2 x + \sin 2 y } { \cos 2 x + \cos 2 y } \equiv \tan ( x + y ) .$$ (b) Hence, show that $$\tan 52.5 ^ { \circ } = \sqrt { 6 } - \sqrt { 3 } - \sqrt { 2 } + 2 .$$