<s>[INST] 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.
 [/INST] $$A+\sqrt{B}=\frac{A+C}{2}+\frac{A-C}{2}+\sqrt{A^2-(A^2-B)}$$
$$={\sqrt{\frac{A+C}{2}}}^2+2\sqrt{\frac{(A+\sqrt{A^2-B})(A-\sqrt{A^2-B})}{4}}+{\sqrt{\frac{A-C}{2}}}^2$$
$$={\sqrt{\frac{A+C}{2}}}^2+2\sqrt{\frac{(A+C)(A-C)}{4}}+{\sqrt{\frac{A-C}{2}}}^2$$
$$=\left(\sqrt{\frac{A+C}{2}}+\sqrt{\frac{A-C}{2}}\right)^2$$
</s>