Question 1005148
By expanding the right-hand side (RHS) we can invoke a lot of trig identities:


*[tex \large RHS = 2(\sin \alpha + \sin \beta + \sin \gamma) + \sin 2\alpha + \sin 2\beta + \sin 2\gamma + 2(\sin(\alpha + \beta) + \sin(\alpha + \gamma) + \sin(\beta + \gamma))]

using the fact that *[tex \large \sin(x+y) = \sin x \cos y + \sin y \cos x].


Since *[tex \large \alpha + \beta + \gamma = 0], we have that *[tex \large \sin(\alpha + \beta) = \sin(-\gamma) = -\sin \gamma], etc., so we can simplify as follows:


*[tex \large RHS = 2(\sin \alpha + \sin \beta + \sin \gamma) + \sin 2\alpha + \sin 2\beta + \sin 2\gamma - 2(\sin \alpha - \sin \beta - \sin \gamma)]


*[tex \large = \sin 2\alpha + \sin 2\beta + \sin 2\gamma]


*[tex \large = LHS]