Double radical proof I'm trying to prove that
$$
\sqrt{A+\sqrt{B}}=\sqrt{\frac{A+C}{2}}+\sqrt{\frac{A-C}{2}}
$$
With
$$
C=\sqrt{A^2 - B}
$$
How can I handle this?
Edit: obviously is easy that this holds when you know the r.h.s., but my question is: how to get the r.h.s. when you only know the l.h.s.

$$A+\sqrt{B}=\frac{A}{2}+\frac{C}{2}+\frac{A}{2}-\frac{C}{2}+\sqrt{B},$$
where $C=\sqrt{A^2-B}$. But
$$\sqrt{B}=\sqrt{A^2-(A^2-B)}=\frac{2}{2}\sqrt{(A+\sqrt{A^2-B})(A-\sqrt{A^2-B})}=2\sqrt{\frac{A+C}{2}}\sqrt{\frac{A-C}{2}}.$$
Then
$$A+\sqrt{B}=\frac{A+C}{2}+\frac{A-C}{2}+2\sqrt{\frac{A+C}{2}}\sqrt{\frac{A-C}{2}}=\left(\sqrt{\frac{A+C}{2}}+\sqrt{\frac{A-C}{2}}\right)^2.$$
Taking square roots gives the answer.
