Power Analysis for Non-sinusoidal excitations
One way to look at this is, when you integrate over time to calculate power, all cross terms with different frequencies integrate to zero, because sinusoidal of different frequencies are orthogonal.

A very simple case as example:
$$\int (v_1 \cos(w_1 t) + v_2 \cos(w_2 t))^2 dt = \int (v_1 \cos(w_1 t))^2 dt + \int (v_2 \cos(w_2 t))^2 dt$$
because
$$\int v_1 v_2 \cos(w_1 t) \cos(w_2 t) dt = 0$$

Therefore calculating the power of each frequency then adding them together is equal to adding the voltages (and currents) together then calculating the power.