# Prove:`sin^(-1)(-x)=-sin^(-1)(x)`

mathsworkmusic | (Level 2) Educator

Posted on

The sine function is an odd function, meaning that it has `````180^o` rotational symmetry about the origin, and in particular that

`sin(-x) = -sin(x)`

To prove that

`sin^(-1)(-x) = -sin^(-1)(x)`    (1)

take the sine of both sides:

`sin[sin^(-1)(-x)] = sin(-sin^(-1)x)`

Since doing the inverse sine operation followed by sine operation results in the identity operation, this implies that

`-x = sin(-sin^(-1)x)`

Since, as noted above, sine is an odd function,  then equation (1) holding true implies that

`-x = - sin(sin^(-1)x) = -x`

Therefore equation (1) must be true. [Taking the derivative of both sides to prove this is  overcomplicating the problem]

Sources:

jeremies | Student, College Freshman | (Level 2) eNoter

Posted on

Never mind, I figured it out. You take the derivative of both sides and remember
that on the left side that your ``,thus, the answer for both sides will be: `-1/sqrt(1-x^2)`