When we have 1D standing waves, we can write them as the sum of two propagating wave in opposite directions that give the formula $sin(kx)cos(wt)$.

When I try to do this for 2D waves (I mean 2D by the fact there are $k_x$ and $k_y$,) I don't have the right expression :

$e^{i(k_x x + k_y y - wt)}+e^{i(k_x x + k_y y + wt)}=e^{i(k_x x + k_y y)}.2cos(wt)$

If I take the imaginary part I will have $sin(k_x x +k_y y)$ and not $sin(k_x x)sin(k_y y)$.

Am I wrong somewhere or we can't say that 2D stationnary waves are the sum of two propagating waves in opposite directions ?

Thank you.