<s>[INST] Why does $\sin(x)$ + $\sin(x+a)$ always come out as a sine function? 
How is it possible mathematically? In short, this says, $\sin(x)$ + $\sin(x+a)$ must be equal to some function like $b$ $\sin(x+c)$ somehow, but how?
 [/INST] The answer lies in the sum-to-product formula for sine,
$$\sin(a) + \sin(b) = 2 \sin \left( \frac{a+b}{2} \right) \cos \left( \frac{a-b}{2} \right)$$
Replace $a$ with $x$ and $b$ with $x+a$ in the above formula to get
$$\begin{align*}
\sin(x) + \sin(x+a)
 &= 2 \sin \left( \frac{x+x+a}{2} \right) \cos \left( \frac{x-(x+a)}{2} \right) \\
 &= 2 \sin \left( x + \frac a 2 \right) \cos \left( \frac a 2 \right) 
\end{align*}$$
Note that $\cos(a/2)$ is a constant, so you really have a sine wave, with amplitude $2 \cos(a/2)$, and shifted left by $a/2$ units.
(Reference sheet for trig identities.)
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