Explanation

(c)
Given equations may be written as
$\cos{x + \cos{y = - \cos\alpha}}$
and$\ \sin{x + \sin{y = - \sin\alpha}}$
$\Rightarrow 2\cos{\left( \frac{x + y}{2} \right)\cos\left( \frac{x - y}{2} \right) = - \cos\alpha}$ …(i)
and$\ 2\sin{\left( \frac{x + y}{2} \right)\cos{\left( \frac{x - y}{2} \right) = - \sin\alpha}}$ …(ii)
From Eqs.(i)and (ii),we get
$\frac{2\cos{\left( \frac{x + y}{2} \right)\cos\left( \frac{x - y}{2} \right)}}{2\sin{\left( \frac{x + y}{2} \right)\cos\left( \frac{x - y}{2} \right)}} = \ \frac{\cos\alpha}{\sin\alpha}$
$\Rightarrow \cot{\left( \frac{x + y}{2} \right) = \cot\alpha}$