Answer
See the steps.
Work Step by Step
$\sin{(\dfrac{\pi}{4}+x)} + \sin{(\dfrac{\pi}{4}-x)} = $
$$\sin{\dfrac{\pi}{4}} \cos{x} + \cos{\dfrac{\pi}{4}} \sin{x} + \sin{\dfrac{\pi}{4}} \cos{x} - \cos{\dfrac{\pi}{4}} \sin{x}$$
$LHS = 2 \sin{\dfrac{\pi}{4}} \cos{x} = 2 \times \dfrac{1}{\sqrt{2}} \cos{x} = \sqrt{2} \cos{x}$
$LHS = RHS$
