<s>[INST] Prove $\sin^2 a + \sin^2 b + \sin^2 c = 2(1-\cos a \cos b \cos c)$ when $a=b+c$ I want to prove this identity:

$$\sin^2 a + \sin^2 b + \sin^2 c = 2(1-\cos a \cos b \cos c) \qquad\text{when}\;a=b+c$$

Can somebody give a hint in the easiest way possible? I am debugging this for hours and can't get the left side to be the right.
 [/INST] Let $\, X := e^{ia}, Y := e^{ib}, Z := e^{ic}. \,$ Now expand and factor
$$ W \!:=\! \sin^2 a \!+\! \sin^2 b \!+\! \sin^2 c \!-\! 2(1 \!-\! \cos a \cos b \cos c) \!=\! 
\frac{(X Y \!-\! Z) (Y Z \!-\! X) (Z X \!-\! Y) (X Y Z \!-\! 1)}{(2 X Y Z)^2}.\,$$
This leads to
  $$\, W = 4 \sin\frac{a+b-c}2 \sin\frac{a-b+c}2 \sin\frac{-a+b+c}2 \sin\frac{a+b+c}2 \,$$
which is zero iff at least one of
 $\, a+b-c,\, a-b+c,\, -a+b+c,\, a+b+c \,$ are multiples of $\,2\pi.\,$
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