I don't know why most of the posts do not even attempt to answer the question?
The question (KristianNorway, post #1) was: "What happens to the induced drag (Cdi) when I lower the flaps, everything else being equal?"
saman (post #5) mostly already answered the question by reminding us of the formula *) for Cdi.
Cdi = k* Cl_squared / pi * AR
Here k is a factor depending on the spanwise lift distribution. For an elliptical lift distribution k=1, in all other cases k>1.
AR is the aspect ratio of the wing.
If we start with basic flaps (simple trailing-edge flaps), and "everything else equal", i.e., speed, height, weight and wing surface, Cl is unchanged, and so is AR.
The only thing that changes is k.
Whether k increases or decreases depends on the change in lift distribution between the clean wing and the wing with the flaps down.
An 'ideal' clean wing would have a near-elliptical lift distribution, but in practice this is not necessarily the case, so whether Cdi increaes or decreases will depend on the design of the wing.
'Complex' flaps, such as Fowler flaps, increase the wing area. However, both Cl and AR will decrease proportionally to S (wing area), so Cdi remains the same for 'basic' and 'complex' flaps.
*) BTW, before somebody contests the use of a simple-looking formula : while the formula is derived from theoretical wing flow analysis, the experimental results are so close to the theoretical ones, that to all practical extent and purposes they are the same.
Most of the other points raised in the various posts, while not necessarily wrong, do not answer the original question. Matters such as total drag, AoA, L/D curves, low or high speed, wing camber, and others, do not "enter into the equation".
CJ