z=x+y, so, differentiating: dz/dx=1+dy/dx. Therefore dy/dx=dz/dx-1 (dx//dx=1).

The question says dy/dx=cos(x+y).

So, substituting: dz/dx-1=cos(z). We can write this: dz-dx=cos(z)dx, and dz=dx(1+cos(z)).

So dz/(1+cos(z))=dx and the variables x and z have been separated.

∫((1/(1+cos(z))dz)=x; cos2A=2cos^2(A)-1 as a trig identity, so 2cos^2(A)=1+cos(2A). Now if 2A=z, A=z/2.

Therefore 1+cos(z)=2cos^2(z/2) and we have ∫(½sec^2(z/2)dz)=tan(z/2).

We can replace z with x+y: tan((x+y)/2)=x+C where c is the constant of integration.

When x=0, y=π/4, tan(π/8)=C and tan((x+y)/2)=x+tan(π/8). This looks like the most concise way to express the equation.

tan((x+y)/2)=tan(x/2+y/2)=(tan(x/2)+tan(y/2))/(1-tan(x/2)tan(y/2)).