Question

How would I do this?

Answer 1

(-1,1)

Explanation:

Let
`\sec(\sin^(-1)(x))=\sec\theta` so
`\theta=\sin^(-1)(x)` and
`\sin\theta=x`:

`\displaystyle \sin\theta=\frac{\text{Opposite}}{\text{Hypotenuse}}=(x)/(1)`

`\displaystyle \sec\theta=(1)/(\cos\theta)=\frac{\text{Hypotenuse}}{\text{Adjacent}}=(1)/(√(1-x^2))`

Adjacent side can be calculated from the Pythagorean Theorem

Since
`1-x^2 > 0`, then
`-1 < x < 1`, making the domain
`(-1,1)` for the composite trig function.