Answer
See the steps.
Work Step by Step
$\sin{(\dfrac{\pi}{6}+x)} = \sin{\dfrac{\pi}{6}} \cos{x} + \cos{\dfrac{\pi}{6}}\sin{x}$
$\sin{(\dfrac{\pi}{6}-x)} = \sin{\dfrac{\pi}{6}} \cos{x} - \cos{\dfrac{\pi}{6}}\sin{x}$
$LHS = \sin{(\dfrac{\pi}{6}+x)} +\sin{(\dfrac{\pi}{6}-x)} = $
$$\sin{\dfrac{\pi}{6}} \cos{x} + \cos{\dfrac{\pi}{6}}\sin{x}+\sin{\dfrac{\pi}{6}} \cos{x} - \cos{\dfrac{\pi}{6}}\sin{x}$$
$LHS = 2 \sin{\dfrac{\pi}{6}} \cos{x} = 2 \times \dfrac{1}{2} \cos{x} = \cos{x}$
$LHS = RHS$
