Answer
The graphs do not appear to coincide.
Work Step by Step
The graphs of $y=\cos (x+ \frac{\pi }{2})$ (the red graph) and $y=\cos x + \cos \frac{\pi }{2}$ (the blue graph) do not coincide.
For $x=\pi$ we have$$\cos (\pi + \frac{\pi }{2})=0 \neq -1= \cos \pi + \cos \frac{\pi }{2}.$$
