Question

If $0<x,y<2\pi$, the number of solutions of the system of the equations $\sin x\sin y=\frac{3}{4}$ and $\cos x\cos y=\frac{1}{4}$ is

Answer 1

From the given equations we have
$\cos x\cos y+\sin x\sin y=1=\cos (x-y)$ ...(1)
$\cos x\cos y-\sin x\sin y=-\frac{1}{2}=\cos (x+y)$ ...(2)
From (1) we get $x-y=0\Rightarrow x=y$
From (2) we get $x+y={120}^0,{240}^0\Rightarrow x=y={60}^0,{120}^0$