Answer
$$ \frac{\sqrt{2}}{2}\left(\sin \theta + \cos \theta\right) $$
Work Step by Step
Since
$$\sin(x+y)= \sin x\cos y+\cos x\sin y $$
Then
\begin{align*}
\sin(\theta+45^{\circ})&= \sin \theta\cos 45^{\circ}+\cos \theta\sin 45^{\circ}\\
&= \frac{\sqrt{2}}{2}\sin \theta + \frac{\sqrt{2}}{2}\cos \theta\\
&= \frac{\sqrt{2}}{2}\left(\sin \theta + \cos \theta\right)
\end{align*}