<s>[INST] A trigonometry question from STEP examination Show that if at least one of the four angles A ± B ± C is a multiple of π, then
$$\sin^4A + \sin^4 B + \sin^4 C − 2 \sin^2 B \sin^2 C − 2 \sin^2 C \sin^2 A
− 2 \sin^2 A \sin^2 B + 4 \sin^2 A \sin^2 B \sin^2 C = 0$$
I want to start with proving $\sin(A+B+C)$ or $(\sin(A)+\sin(B)+\sin(C))^2$, however, I failed in both cases.
 [/INST] Hint:
First of all writing $\sin A=a$ etc.,
$$a^4+b^4+c^4-2a^2b^2-2b^2c^2-2c^2a^2=(a^2+b^2-c^2)^2-(2ab)^2$$
$$=(a+b+c)(a+b-c)(a-b+c)(a-b-c)$$
Now if $A+B+C=\pi$
by this $\sin A+\sin B+\sin C=4\cos\dfrac A2\cos\dfrac B2\cos\dfrac C2$
and by this $\sin A+\sin B-\sin C=4\sin\dfrac A2\sin\dfrac B2\cos\dfrac C2$
Use $\sin2x=2\sin x\cos x$
We shall same expressions in some order if $A\pm B\pm C=\pi$
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