Explanation

(b)
Since, $5\cos{x + 12\cos{y = 13}}$
$\Rightarrow \ \ \left( 5\cos{x + 12\cos y} \right)^{2} + \left( 5\sin{x + 12\sin y} \right)^{2}$
$= (13)^{2} + \left( 5\sin{x + 12\sin y} \right)^{2}$
$\Rightarrow 25 + 144 + 120\left( \sin{x\sin{y + \cos{x\cos y}}} \right)$
$= 169 + \left( 5\sin{x + 12\sin y} \right)^{2}$
$\Rightarrow \ \ \left( 5\sin{x + 12\sin y} \right)^{2} = 120\cos(x - y)$
$\because\ - 1 \leq \cos(x - y) \leq 1$
$\Rightarrow \ - 120 \leq 120\cos{(x - y)} \leq 120$
$\therefore$ Maximum value of $5\sin{x + 12\sin{y = \sqrt{120}}}$