Explanation

(c)
$\cos{2A + \cos{2B + \cos{2C}}}$
$= 2\cos{(A + B)\cos{(A - B) + 1 - 2\sin^{2}C}}$
$= 2\ cos\left( \frac{3\pi}{2} - C \right)\cos{(A - B) + 1 - 2\sin^{2}C}$
$\left\lbrack \because A + B + C = 270{^\circ}\ \Rightarrow \ \ B + A = \frac{3\pi}{2} - C \right\rbrack$
$= 1 - 2\sin{C\lbrack\cos{(A - B) + \sin C}\rbrack}$
$= 1 - 2\sin{C\lbrack\cos{(A - B) - \cos{(A + B)}}\rbrack}$
$= 1 - 4\sin{A\sin{B\sin C}}$