In physics problems involving 2 wave sources on axis, the extrema of an equation of the form

$$y = \sin x + \sin(x + a)$$

have significance as related to the amplitude of the superposition of the two wave sources. In order to solve for these, I have personally used numerical methods. I have also seen that:

$$\sin x + \sin(x + a) = 2 \cos(a/2) \sin(x + a/2)$$

can be used to make these calculations easier. The truth of this can be verified easily with wolfram alpha or a graphing calculator, but what is the easiest way to prove such an identity?