# How do you simplify sin(2sin^-1(10x))?

Oct 28, 2017

$\sin \left(2 {\sin}^{- 1} 10 x\right) = 20 x \sqrt{1 - 100 {x}^{2}}$

#### Explanation:

Let ${\sin}^{- 1} 10 x = y$

then $\sin y = 10 x$ and $\cos y = \sqrt{1 - {\left(10 x\right)}^{2}} = \sqrt{1 - 100 {x}^{2}}$

and $\sin \left(2 {\sin}^{- 1} 10 x\right)$

= $\sin \left(2 y\right)$

= $2 \sin y \cos y$

= $2 \times 10 x \times \sqrt{1 - 100 {x}^{2}}$

= $20 x \sqrt{1 - 100 {x}^{2}}$