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