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Simplify cos(2sin^-1(x))
(sin^-1 = sin inverse)
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They want you calculate cos(2a) given
sin(a) = x. (1)
Use the formula for cos(2a) via sin(a)
cos(2a) = cos^2(a) - sin^2(a) = 1 - 2sin^2(a).
Substitute here sin(a) = x based on (1). You will get
cos(2sin^-1(x)) = 1 - 2x^2.
ANSWER. cos(2sin^-1(x)) = 1 - 2x^2, -1 <= x <= 1.
Solved, with full explanations.