sin(x+¼π)=sin(x)cos(¼π)+cos(x)sin(¼π).
But sin(¼π)=cos(¼π), because sin(¼π)/cos(¼π)=tan(¼π)=1.
sin(¼π)=cos(¼π)=√2/2, so sin(x+¼π)=(√2/2)(sin(x)+cos(x)) or (sin(x)+cos(x))/√2.
This means that, if we plot √2sin(x+¼π) we get the graph of sin(x)+cos(x).
This further implies that we have a simple sine wave with a maximum value of √2 instead of 1 and a minimum value of -√2 instead of -1 (that is, amplitude=√2 instead of 1). And there's a phase shift of -¼π (that is, -45°) which pushes the sine wave to the left by one eighth of a cycle.