# How do you simplify sin (2 * arcsin (x))?

Nov 8, 2016

The answer is $= 2 x \sqrt{1 - {x}^{2}}$

#### Explanation:

Let $y = \arcsin x$, then $x = \sin y$

$\sin \left(2 \arcsin x\right) = \sin 2 y = 2 \sin y \cos y$

${\cos}^{2} y + {\sin}^{2} y = 1$

${\cos}^{2} y = 1 - {x}^{2}$$\implies$$\cos y = \sqrt{1 - {x}^{2}}$

$\therefore \sin \left(2 \arcsin x\right) = 2 x \sqrt{1 - {x}^{2}}$

Jun 15, 2018

If we interpret $\arcsin a$ as all the solutions to $\sin x = a$ then
$\sin \left(2 \arcsin x\right) = 2 \left(\sin \arcsin x\right) \left(\cos \arcsin x\right) = 2 x \cos \arcsin \left(\frac{x}{1}\right) = \pm 2 x \sqrt{1 - {x}^{2}}$
$\arcsin \left(\frac{x}{1}\right)$ refers to a right triangle, opposite $x$, hypotenuse $1$ so adjacent $\sqrt{1 - {x}^{2}}$. The sign is ambiguous so we prepend $\pm$.