The value of `sin^(-1)[xsqrt(1-x)-sqrt(x)sqrt(1-x^2)]` is equal to `sin^(-1)x+sin^(-1)sqrt(x)` `sin^(-1)x-sin^(-1)sqrt(x)` `sin^(-1)sqrt(x)-sin^(-1)x` none of these

Options

A. `sin^(-1) x + sin^(-1) sqrtx`
B. `sin^(-1) x - sin^(-1) sqrtx`
C. `sin^(-1) sqrtx - sin^(-1) x`
D. none of these

Answer

B

Explanation

Let `x = sin theta and sqrtx = sin phi, " where " x in [0, 1]`
`rArr theta, phi in [0, pi//2]`
`rArr theta - phi in [(-pi)/(2), (pi)/(2)]`
Now, `sin^(-1) (x sqrt(1 -x) - sqrtx sqrt(1 - x^(2)))`
`= sin^(-1) (sin theta sqrt(1 - sin^(2) phi) - sin phi sqrt(1 - sin^(2) theta))`
`= sin^(-1) (sin theta cos phi - sin phi cos theta)`
`= sin^(-1) sin (theta - phi) = theta - phi`
`= sin^(-1) (x) - sin^(-1) (sqrtx)`