Why is $\cos(x/2)+2\sin(x/2)=\sqrt5 \sin(x/2+\tan^{-1}(1/2))$ true? According to Wolfram Alpha the following equality holds:
$$\cos\left(\frac{x}{2}\right)+2\sin\left(\frac{x}{2}\right)=\sqrt5 \sin\left(\frac{x}{2}+\tan^{-1}\left(\frac{1}{2}\right)\right)$$
I also checked it numerically. Why is it true?

Using $\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$,
\begin{align}
  \sin\left(x + \tan^{-1} \left(\frac 1 2\right) \right)
  &= \sin x \cos \left(\tan^{-1}\left(\frac 1 2\right)\right) +
     \cos x \sin \left(\tan^{-1}\left(\frac 1 2\right)\right) \\
  &= \sin x \frac 2 {\sqrt 5} + \cos x \frac 1 {\sqrt 5}.
\end{align}
