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