Answer
The graphs do not appear to coincide.
Work Step by Step
The graph of $y=\sin (x+ \frac{\pi }{4})$ (the red graph) and $y=\sin x+ \frac{\pi }{4}$ (the blue graph) do not coincide.
For $x=\frac{\pi }{4}$ we have$$\sin (\frac{\pi }{4} + \frac{\pi }{4})=1 \neq \frac{2\sqrt{2}+\pi }{4} = \sin \frac{\pi }{4} + \frac{\pi }{4}.$$
