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:
$$\sin\left(\frac{\pi}{2}-x\right)=\sin(x+y)=\sin\left(\frac{\pi}{2}-y\right).$$

Hint:
$$\frac{df}{dx}=\cos x - \sin(x+y) = 0$$
$$\frac{df}{dy}=\cos y - \sin(x+y) = 0$$
Exploding the $\sin$s
$$\cos x - \sin x \cos y - \cos x \sin y = 0$$
$$\cos y - \sin x \cos y - \cos x \sin y = 0$$
Dividing the first with $\cos x$, the second with $\cos y$:
$$1 - \tan x \cos y - \sin y = 0$$
$$1 - \sin x - \tan x \cos y = 0$$
Expressing $\tan x$ from the first:
$$\tan x = \frac{1- \sin y}{\cos y}$$
...and now substitute it into the second equation. You have to express $\sin x$ with $\tan x$, there is a trigonometrical identity for that, but unfortunately this small margin is too narrow here to contain. ;-)
But the result will be a second degree equation and you will simply solve it.
