How did they get this result. It does not make sense, can anybody show me how they derived this result. My question is how did they totally remove e^(jkwot), by what identity and I know it is modulus so it has to be squared, but I tried several time, I just do not how they came to this result.
Because $$|a_ke^{jk\omega_0 t}|^2=|a_k|^2\,|e^{jk\omega_0 t}|^2=|a_k|^2 \cdot 1.$$
Just notice from the polar form of a complex number, $re^{j\theta}$, that its magnitude is $r$.