How to integrate this?

1. The problem statement, all variables and given/known data
How to integrate the double integral cos(x+y) dy dx from 0 to pi and from 0 to pi again.

2. Relevant equations
The answer is -4.

3. The attempt at a solution
Here's the work:
u=x+y
du=dy
cos(u)du=sin(x+y)
integral of [sin(x+y)] evaluate from 0 to pi dx from 0 to pi
integral of (sin(x+pi)-sin(x))dx from 0 to pi
And what's next?

 

But I can't. Because sin(x+pi)-sin(x)=sin(x)+sin(pi)-sin(x)=sin(pi)=0. The integral of 0 is?

 

Then what's sin(a+b)?

 

@Fightfish , thank you so much for the info! I got it!