<s>[INST] Prove that $ABC$ is right-angled Prove that if $\cos^2{A} + \cos^2{B} + \cos^2{C} = 1$, then $ABC$ is right-angled.
I only found that $\sin^2{A} + \sin^2{B} + \sin^2{C} = 2$, but I have no idea what to do next.
Thank you in advance for your answers!
 [/INST] As you found that $\sin^2{A} + \sin^2{B} + \sin^2{C} = 2$ note that $\sin^2 \theta + \cos^2 \theta =1$
Then $1-\cos^2 A + 1- \cos^2 B + 1 - \cos^2 C = 2$
$ 3 -(\cos^2 A + \cos^2 B + \cos^2 C)= 2$
$3 - 2 = \cos^2 A + \cos^2 B + \cos^2 C$
$\cos^2 A + \cos^2 B + \cos^2 C =1$
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