He doesn't use any formulas at all. The only thing he is doing is changing the place of the brackets.
Hint: Start with eliminating the brackets in the term (L - 2*Ls)*Ls*sin(beta).
You can set them equal to each other as they are both equal to 0 (it works as well with every other number, variabele, and the like). Moreover your method should work fine, though you can only divide by 2cos((θ1+θ2)/2) as the cos is not equal to 0 (So you should find more solutions).
Though...