SOLUTION: Simplify cos(2sin^-1(x)) (sin^-1 = sin inverse)

Question 1200038: Simplify cos(2sin^-1(x))
(sin^-1 = sin inverse)
Answer by ikleyn(52835) (Show Source): You can put this solution on YOUR website!
.
Simplify cos(2sin^-1(x))
(sin^-1 = sin inverse)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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.

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