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# 3 arccos x = arccos (4x^3 -3 x )#
Sometimes trig is less about doing math and more about recognizing math when we see it. Here we recognize #4x^3 -3x# as the cosine triple angle formula, #\cos(3 \theta)# when #x=\cos \theta#.
Factoid: #4x^3-3x# is also called #T_3(x)#, the third Chebyshev Polynomial of the first kind. In general, #\cos(nx) = T_n(\cos x).#
We'll assume #arccos# refers to the principal value. I prefer to call the principal #text{Arc}text{cos}# but that's harder to type.
Enough background. Once we've recognized the triple angle formula the proof is easy.
Proof :
Let #theta = arccos x.#
#x=cos theta#
# cos 3 theta = 4 cos^3 theta - 3 cos theta #
# cos 3 (arccos x) = 4x^3 - 3 x #
# 3 arccos x = arccos(4x^3 - 3x) quad sqrt#
