Chapter 7 - Section 7.3 - Double-Angle, Half-Angle, and Product-Sum Formulas - 7.3 Exercises - Page 562: 103

See explanation

Let $u = arcsin(x$) and $x = \sin(u)$, then $2u = arccos(1-2x^2)$ $\cos(2u)=1-2x^2$ Use Cosine double angle identity: $\cos(2u)=1-2\sin^2(u)$ $1-2\sin^2(u)=1-2x^2$ Since $x = \sin(u)$ then, $1-2\sin^2(u)=1-2\sin^2(u)$, Proving the identity