Explanation

(a)
$\cos^{2}{A + \cos^{2}{B + \cos^{2}C}}$
$= \frac{1 + \cos{2A}}{2} + \frac{1 + \cos{2B}}{2} + \cos^{2}C$
$= 1 + \frac{1}{2}\left( \cos{2A + \cos{2B}} \right) + \cos^{2}C$
$= 1 + \frac{2}{2}\left\lbrack \cos{(A + B)\cos(A - B)} \right\rbrack + \cos^{2}C$
$= 1 + \cos{C\cos{(A - B) + \cos C\cos{(A + B)}}}$
$= 1 + \cos{C\lbrack\cos{(A - B) + \cos{(A + B)}}\rbrack}$
$= 1 + 2\cos{C\cos{B\cos A}}$
$\Rightarrow \cos^{2}{A + \cos^{2}{B + \cos^{2}{C - 2\cos{A\cos{B\cos{C = 1}}}}}}$