Expert Answers
mathsworkmusic eNotes educator| Certified Educator

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]

nick-teal eNotes educator| Certified Educator

When proving things are equal, you are actually not allowed to take the derivative.  The way that mathsworkmusic did this, taking the sin of both sides, is the proper way to solve this problem.

jeremies | Student

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)`