Explanation

(c)
$3\left( \sin{x - \cos x} \right)^{4} + 6\left( \sin{x + \cos x} \right)^{2} + 4\left( \sin^{6}{x + \cos^{6}x} \right)$
$= 3\left( 1 - \sin{2x} \right)^{2} + 6\left( 1 + \sin{2x} \right) + 4\left\{ \left( \sin^{2}{x + \cos^{2}x} \right)^{3} - 3\sin^{2}{x\cos^{2}{x \bullet \left( \sin^{2}{x + \cos^{2}x} \right)}} \right\}$
$= 3\left( 1 - 2\sin{2x + \sin^{2}{2x}} \right) + 6 + 6\sin{2x + 4\left\{ 1 - 3\sin^{2}{x\cos^{2}x} \right\}}$
$= 3\left\{ 1 - 2\sin{2x + \sin^{2}{2x + 2 + 2\sin{2x}}} \right\} + 4\left\{ 1 - \frac{3}{4}\sin^{2}{2x} \right\}$
$= 13 + 3\sin^{2}{2x - 3\sin^{2}{2x = 13}}$