Solution: f (x) = 2cos2x+2sinxcosx = cos2x+1+sin2x = radical 2sin(2x+π/4)+ 1.

F(α)=f(β), so (2α+π/4)+(2β+π/4)=π, α+β=π/4.

So tan(α+β)= 1.

Because sinα=sin(π-α), if the sine values of two angles are equal, then the two angles are equal or the sum of the two angles is π.

Sin(2α+π/4)=sin(2β+π/4), and the two angles are not equal, so the sum of the two angles is equal to π.

If you don't understand, you can keep asking.