<s>[INST] Finding critical points of $f(x,y)= \sin x+\sin y + \cos(x+y)$ 
Find the critical points of function$$
f(x,y)=\sin x + \sin y + \cos(x+y),$$
  where $0<x<\dfrac{\pi}{2}$, $0<y<\dfrac{\pi}{2}$.

What I have done:
$$f_{x}=\cos(x)-\sin(x+y),\\
f_{y}=\cos(y)-\sin(x+y).$$
From $f_{x}=0$, $\cos(x)=\sin(x+y)$. From $f_{x}=0$, $\cos(y)=\sin(x+y)$. I do not know where to go from here.
My attemps:
$$\sin\left(\frac{\pi}{2}-x\right)=\sin(x+y)=\sin\left(\frac{\pi}{2}-y\right).$$
 [/INST] $$f_x = \cos x - \sin(x+y) = 0\\
f_y = \cos y - \sin(x+y) = 0$$
Subtracting one from the other we get $\cos x = \cos y$ and with the restrictions of $x,y$ to $(0,\frac {\pi}{2})$ we can say $x= y$ and $\cos x - \sin 2x = 0$
$$\cos x(1-\sin x) = 0\\x = \frac {\pi}{6}$$  
$\frac {\pi}{2}$ would also solve the equation but the domain says strictly less than $\frac{\pi}{2}$
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