<s>[INST] Is this true that $(\cos^2A+\cos^2B+\cos^2C+2\cos A\cos B\cos C=1 \implies A+B+C=\pi)$? Assume that  $A,B,C$ are positive real numbers and $A,B,C \in (0,\frac{\pi}{2}]$ and we have
$$\cos^2A+\cos^2B+\cos^2C+2\cos A\cos B\cos C = 1 $$
prove or disprove that $$A+B+C=\pi$$
 [/INST] With an obvious shorthand notation, we can solve the equation for $\cos C$:
$$c^2+2abc+a^2+b^2-1=0,$$
$$c=-ab\pm\sqrt{a^2b^2-a^2-b^2+1}=-ab\pm\sqrt{(1-a^2)(1-b^2)}.$$
Then
$$\cos C=-\cos A\cos B\pm \sin A\sin B=-\cos(A\pm B)$$
and
$$\pm C=\pi\pm A\pm B.$$
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