Answer
The graphs do not appear to coincide.
Work Step by Step
The graph of $y=\cos (x+ \frac{\pi }{4})$ (the red graph) and $y=\cos x+ \frac{\pi }{4}$ (the blue graph) do not coincide.
For $x=\frac{\pi }{4}$ we have$$\cos (\frac{\pi }{4} + \frac{\pi }{4})=0 \neq \frac{2\sqrt{2}+\pi }{4} = \cos \frac{\pi }{4} + \frac{\pi }{4}.$$
