Question

Find an algebraic expression equivalent to the given expression.

Answer 1

sin(sin^{-1}(x))=x

Explanation:

an alternative method would be to use the fact that

`f( {f}^( - 1) (x)) = x`

what this is saying is that if we apply a function on its inverse, we will get back our original input.

by definition, f^{-1} or the inverse of f will undo whatever f does, and vice versa.

think of it this way: whatever f(x) does, its inverse f^{-1}(x) undoes. so whatever f^{-1}(x) does, f(x) undoes.

so, we are applying f^{-1} onto x, and then f will undo whatever it does, and our result will be x.

thus, letting f(x) = sin(x),

we have

`\sin( { \sin(x) }^( - 1) ) = x`

note that x can only take values from [-1, 1], because of the domain of sin(x).