Step 1:
$\cos y=x\cos(a+y)$
$x=\Large\frac{\cos y}{\cos(a+y)}$
$\Large\frac{dx}{dy}=\frac{\cos(a+y).\Large\frac{d}{dy}(\cos y)-\cos y.\Large\frac{d}{dy}(\cos(a+y))}{\cos^2(a+y)}$
$\quad\;=\Large\frac{\cos(a+y).(-\sin y)-\cos y.(-\sin(a+y))}{\cos^2(a+y)}$
$\quad\;=\Large\frac{\sin(a+y)\cos y-\cos(a+y)\sin y}{\cos^2(a+y)}$
$\quad\;=\Large\frac{\sin(a+y-y)}{\cos^2(a+y)}$
$\quad\;=\Large\frac{\sin a}{\cos^2(a+y)}$
Step 2:
$\Large\frac{dx}{dy}=\frac{\sin a}{\cos^2(a+y)}$
$\Large\frac{dy}{dx}=\frac{\cos^2(a+y)}{\sin a}$
Hence proved.