Explanation

(a)
$\because\ \sin{x + \sin^{2}{x = 1\ \ \Rightarrow}}\sin{x = \cos^{2}x}$
Now, $\cos^{12}{x + 3\cos^{10}{x + 3\cos^{8}{x + \cos^{6}x}}}$
$= \sin^{6}{x + 3\sin^{5}{x + 3\sin^{4}{x + \sin^{3}x}}}$
$= \left( \sin^{2}{x + \sin x} \right)^{3} = 1$