How does causality (i.e. unit step) affect the DTFT of a sine or cosine wave?
Multiplication in the time domain is convolution in the frequency domain. The frequency response of a unit step has 1/frequency shape and it's pretty messy. 

What you will see is that the overall frequency response is not a delta impulse anymore but that the lines widens and that the get "skirts". These will bigger and more "wiggly" for the cosine than for the sine as the step function slices right through the maximum of the cosine.