simplify the following expression sin(v+x)-sin(v-x)?

Question

Rewrite the expression as an algebraic expression in x. sin(2cos^-1x)

Answer 1

sin(v+x) = sinv*cosx + cosv*sinx
sin(v-x) = sinv*cosx - cosv*sinx
subtract to get
2cosv*sinx

sin(cos^-1 x) = √(1-x^2)
by definition, cos(cos^-1 x) = x

sin(2cos^-1 x) = 2sin(cos^-1 x)cos(cos^-1 x)
= 2√(1-x^2) * x
= 2x√(1-x^2)