Simplify the following \( \sin \left(\theta-45^{\circ}\right)+\sin \left(\theta+45^{\circ}\right) \)

To simplify the expression \( \sin \left(\theta-45^{\circ}\right)+\sin \left(\theta+45^{\circ}\right) \), we can use the sine addition formula: \[ \sin(a + b) + \sin(a - b) = 2 \sin(a) \cos(b) \] In our case, let \( a = \theta \) and \( b = 45^{\circ} \). Applying the formula, we get: \[ \sin \left(\theta - 45^{\circ}\right) + \sin \left(\theta + 45^{\circ}\right) = 2 \sin(\theta) \cos(45^{\circ}) \] Since \( \cos(45^{\circ}) = \frac{\sqrt{2}}{2} \), our expression becomes: \[ 2 \sin(\theta) \cdot \frac{\sqrt{2}}{2} = \sqrt{2} \sin(\theta) \] Thus, the simplified form is: \[ \sqrt{2} \sin(\theta) \]