Simple trigonometrical equations

Question

I'm having difficulties in solving the simultaneous equations $$ \begin{cases} \sin(x+y)=\frac{1}{\sqrt{2}}\\ \cos(2x+y)=\frac12 \end{cases} $$ for $0^{\circ}\le x,y\le 90^{\circ}$.

The answer is $x=15^{\circ}$ degrees and $y= 30^{\circ}$.

Answer 1

$x+y=45$
$2x+y=60$
Solving, we get $x=15, y=30$

Answer 2

$$\sin(x+y)=\dfrac1{\sqrt2}=\sin45^\circ$$

$x+y=n180^\circ+(-1)^n45^\circ$ where $n$ is any integer

As $0^\circ<x,y<90^\circ,0^\circ<x+y<180^\circ\implies x+y=45^\circ\ \ \ \ (1)$ or $x+y=135^\circ\ \ \ \ (2)$

Similarly, $2x+y=360^\circ m\pm 60^\circ$ where $m$ is any integer

As $0^\circ<2x+y<270^\circ,2x+y=60^\circ\ \ \ \ (3)$

Solve $(1),(3)$ and $(2),(3)$