Question

The number of solution(s) of the system of equations $x+y=\frac{2\pi }{3}$ and $\cos x+\cos y=\frac{3}{2},$ where $x$ and $y$ are real, is

Answer 1

We have $\cos x+\cos y=\frac{3}{2}$
$\Rightarrow 2\cos \left(\frac{x+y}{2}\right)\cos \left(\frac{x-y}{2}\right)=\frac{3}{2}$
$\Rightarrow 2\cos \frac{\pi }{3}\cos \left(\frac{x-y}{2}\right)=\frac{3}{2}(\text{using:}x+y=\frac{2\pi }{3})$
$\Rightarrow \cos \left(\frac{x-y}{2}\right)=\frac{3}{2},$ which is not possible.
Hence, the system of equaitons has no solution.