Sin(2sin^-1x)=?

1 Answer
Jun 12, 2018

sin(2arcsinx)=2x1x2

Explanation:

Knowing that:

sin(2α)=2sinαcosα

we have:

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

Now, by definition:

sin(arcsinx)=x

Let y=arcsinx, so that x=siny with y(π2,π2) and cosy>0. Then:

cosy=1sin2y=1x2

Finally:

sin(2arcsinx)=2x1x2