Explanation

(a)
$a\cos{2x + b\sin{2x}}$
$= a.\frac{1 - \tan^{2}x}{1 + \tan^{2}x} + b.\frac{2\tan x}{1 + \tan^{2}x}$
$= a.\frac{1 - \frac{b^{2}}{a^{2}}}{1 + \frac{b^{2}}{a^{2}}} + b.\frac{2.\frac{b}{a}}{1 + \frac{b^{2}}{a^{2}}}\ \ \ \ \ \left\lbrack \because\ \tan{x = \frac{b}{a}} \right\rbrack$
$= \frac{a(a^{2} - b^{2})}{(a^{2} + b^{2})} + \frac{2b^{2}a}{a^{2} + b^{2}}$
$= \frac{a^{3} - ab^{2} + 2ab^{2}}{a^{2} + b^{2}} = \frac{a^{3} + ab^{2}}{a^{2} + b^{2}} = a$