I am seeing some questions in my textbook involving phasor transformation with multiple functions of cosine or sine in multiplication with each other but they didn't exactly showed how to do it and I feel like I'm doing something wrong, take a look at this for example:$$H=H_0cos(k_xx)cos(wt-{\beta}z){u_z}$$
(Note: $k_x$ and $\beta$ are real valued and question wants me to find $H$ in the phasor domain.)
My first instinct was to write both cosine functions in the form of $H_0e^{jk_xx}e^{-j{\beta}z}$ then combine the powers of $e$ into one but reverse transformation is not the same function anymore, am I not suppose to apply it to the $cos(k_xx)$ since it is already not in the time domain, like how we do with the attenuation constant $e^{{\alpha}k}$?