Calculating Clebsh-Gordan coefficients I'm reading about Clebsh-Gordan coefficients, and one of the sources gives such an equation to calculate coefficients for state $m=j_1+j_2-1$
$$|j_1+j_2,j_1+j_2-1\rangle\approx J_-|j_1,j_2\rangle=\sqrt{\frac{j_2}{j_1+j_2}}|j_1,j_2-1\rangle+\sqrt{\frac{j_1}{j_1+j_2}}|j_1-1,j_2\rangle$$
so that coefficients are 
$$\langle j_1+j_2,j_1+j_2-1|j_1,j_2-1\rangle=\sqrt{\frac{j_2}{j_1+j_2}}$$
$$\langle j_1+j_2,j_1+j_2-1|j_1-1,j_2\rangle=\sqrt{\frac{j_1}{j_1+j_2}}$$
However, I am not sure how it was derived from $J_-|j_1,j_2\rangle$.
 A: The situation is simplest if one writes
$$
\vert j_1+j_2,j_1+j_2\rangle = \vert j_1j_1\rangle \vert j_2j_2\rangle \tag{1}
$$
where $\vert j_km_k\rangle$ is an angular momentum state for particle $k$.  In this notation the total angular momentum operators as sums 
$J_{x}=J_{1x}+J_{2x}$ etc where $J_{1x}$ acts only on states of the first particle, and $J_{2x}$ acts only on the states of the second particle.   More formally, one should have $J_x=J_{1x}\otimes \hat 1 + \hat 1\otimes J_{2x}$.  
Thus, the ladder operators are $J_\pm = J_{1\pm}+J_{2\pm}$ and one can verify that the state in (1) is killed by $J_+$ and is an eigenstate of $J_z$ with eigenvalue $j_1+j_2$, identifying this (up to a phase which is chosen to be +1) as $\vert jj\rangle$ with $j=j_1+j_2$.
To get the state $\vert j_1+j_2,j_1+j_2-1\rangle $, ladder down from $\vert j_1+j_2,j_1+j_2\rangle$:
\begin{align}
J_-\vert j_1+j_2,j_1+j_2\rangle &= \sqrt{2(j_1+j_2)}\vert j_1+j_2,j_1+j_2-1\rangle \\
&= \left[J_{1-}\vert j_1j_1\rangle\right]\vert j_2j_2\rangle 
+ \vert j_1j_1\rangle \left[J_{2-}\vert j_2j_2\rangle\right]\, ,\\
&=\sqrt{2j_1}\vert j_1,j_1-1\rangle\vert j_2j_2\rangle 
+\sqrt{2j_2}\vert j_1j_1\rangle \vert j_2j_2-1\rangle
\end{align}
from which your expressions follows.
Note that the (unnormalized) state
$\sqrt{2j_2}\vert j_1,j_1-1\rangle\vert j_2j_2\rangle - \sqrt{2j_1}
\vert j_1j_1\rangle \vert j_2,j_2-1\rangle$ is killed by $J_+$ and an eigenstate of $J_z$ with eigenvalue $j_1+j_2-1$, and must be thus proportional to the state $\vert jj\rangle$ with $j=j_1+j_2-1$.
