Question

Find all pairs of $x,y$ that satisfy the equation
$cosx+cosy+cos(x+y)=-3/2$

Answer 1

$\cos x+\cos y-\cos (x+y)=\frac{3}{2}$
$\Rightarrow 2\cos \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right)-2{\cos }^2\left(\frac{x+y}{2}\right)+1=\frac{3}{2}$
$\Rightarrow 4{\cos }^2\left(\frac{x+y}{2}\right)-4\cos \left(\frac{x-y}{2}\right)\cos \left(\frac{x+y}{2}\right)+1=0$
$\Rightarrow {(2\cos \left(\frac{x+y}{2}\right)-\cos \left(\frac{x-y}{2}\right))}^2+{\sin }^2\left(\frac{x-y}{2}\right)=0$
$\Rightarrow 2\cos \left(\frac{x+y}{2}\right)-\cos \left(\frac{x-y}{2}\right)$ or $\sin \left(\frac{x-y}{2}\right)=0$
$\Rightarrow x-y=4n\pi$ or $x+y=4m\pi \pm \frac{2\pi }{3}$
$\Rightarrow x-y=2(2n+1)\pi$ or $x+y=4m\pi \pm \frac{4\pi }{3}$
$\Rightarrow x=2(m+n)\pi \pm \frac{\pi }{3}$ or $y=2(m-n)\pi \pm \frac{\pi }{3}$
$\Rightarrow x=(2m+2n+1)\pi \pm \frac{2\pi }{3}$ or $y=(2m-2n-1)\pi \pm \frac{2\pi }{3}$