<s>[INST] How to prove this trig identity? If $A+B+C=\pi$ then prove:$$\sin^2A+\sin^2B+\sin^2C=2-2\cos A\cos B\cos C$$
I am completely lost on this, please help.
 [/INST] $A+B+C = \pi \implies A + B = \pi - C \implies \sin(A+B)=\sin(C) \land \cos(A+B)=-\cos(C)$
$\sin^2(A) + \sin^2(B) + \sin^2(C) \\ = \sin^2(A) + 1-\cos^2(B) + 1 − \cos^2(A+B) \\ =2+(\sin^2(A)-\cos^2(B))-\cos^2(A+B) \\ = 2+\cos(B-A)\cos(A+B)-\cos(A+B)\cos(A+B) \\ = 2+(\cos(B-A)+\cos(A+B))\cos(A+B) \\ = 2 - 2\cos(A)\cos(B)\cos(C)$
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