Sin(2sin^-1x)=?

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Answer 1 · 2018-06-12T08:27:25

#sin(2arcsinx) = 2xsqrt(1-x^2)#

Knowing that:

#sin(2alpha) = 2sin alpha cos alpha#

we have:

#sin(2arcsinx) = 2sin(arcsinx)cos (arcsinx)#

Now, by definition:

#sin(arcsinx) = x#

Let #y = arcsinx#, so that #x = siny# with #y in (-pi/2,pi/2)# and #cos y >0#. Then:

#cosy = sqrt(1-sin^2y) = sqrt(1-x^2)#

Finally:

#sin(2arcsinx) = 2xsqrt(1-x^2)#