The intensities should be equal no matter how a wave is represented. So clearly I think i'm making some elementary mistake, it seems they are not same :
$$ \Phi(x,t) = A_0 \cos{(kx-wt)} \\ \Phi(x,t) = A_o e^{i(kx-wt)} \qquad \text{complex notation} $$
So in the above it is understood that the physical part of the wave is the real part, and the imaginary part is to simply calculations by avoiding complicated trigonometric calculations of the former representation.
But then intensity is given by square of the amplitude and modulus square of the wavefunction respectively for the two forms. Then, $$ I = \Phi(x,t)^2 = A_0^2 \cos^2{(kx-wt)} \hspace{2cm} \text{for the 'real' representation} \\ \text{while,} \hspace{1cm} I = |\Phi(x,t)|^2 = \Phi^*\Phi = A_0^2.\hspace{2cm} \text{for the complex representation} $$
These are different...What's the error that I am making? My prof couldn't clear it up, gave some explanation I couldn't understand.
Thank you for any answers.