Prove that cos-¹(1-2x²)=2sin-²x?

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Answer 1 · 2018-05-04T11:02:28

# cos(2 arcsin x) = 1 - 2 sin ^2 (arcsin x) = 1 - 2 x^2 quad # so

#2 arcsin x = arccos(1-2x^2)#

#sin ^{-2} x# ?

Presumably that's #sin ^{-1}(x)#. I prefer #arccos# and #arcsin# anyway.

Show #arccos(1-2x^2)= 2 arcsin x#

Let #theta = arcsin x# so #x = sin theta.#

# cos(2 theta) = 1 - 2 sin ^2 theta = 1 - 2 x^2 #

# 2 theta = arccos(1 - 2x^2)#

# 2 arcsin x = arccos(1 - 2x^2)#

This isn't quite as simple as it looks as arcsin and arccos are multivalued.