# 2 arcsin(x) = arcsin(2x$$\sqrt{1 - x^{2}}$$)

We will learn how to prove the property of the inverse trigonometric function 2 arcsin(x) = arcsin(2x$$\sqrt{1 - x^{2}}$$) or, 2 sin$$^{-1}$$ x = sin$$^{-1}$$ (2x$$\sqrt{1 - x^{2}}$$).

Proof:

Let, sin$$^{-1}$$ x = α

Therefore, sin α = x

Now, sin 2α = 2 sin α cos α

sin 2α = 2 sin α $$\sqrt{1 - sin^{2}α}$$

sin 2α = 2x . $$\sqrt{1 - x^{2}}$$

sin 2α =  2x$$\sqrt{1 - x^{2}}$$

Therefore, 2α = sin$$^{-1}$$ (2x$$\sqrt{1 - x^{2}}$$)

2 sin$$^{-1}$$ x = sin$$^{-1}$$ (2x$$\sqrt{1 - x^{2}}$$).

or, 2 arcsin(x) = arcsin(2x$$\sqrt{1 - x^{2}}$$)             Proved

Inverse Trigonometric Functions