If `0< x< 1`,then `tan^(-1)(sqrt(1-x^2)/(1+x))` is equal to

Options

A. `1/2 cos^(-1)x`
B. `cso^(-1)sqrt(1+x)/(2)`
C. `sin^(-1)sqrt(1-x)/(2)`
D. all the above

Explanation

Let `x = cos theta` Then
`0ltxlt1 rarr 0ltcos theta lt1 rarr0lttheta lt(pi)/(2)`
Now
`tan^(-1)sqrt(1-x^(2))/(1+x)`
`tan^(-1)sqrt(1-x^(2)/(1+x)`
Thus option (a)is true
`cos^(-1)sqrt(1+x)//(2)=cos^(-1)(cos(theta)/(2))=1/2theta=1/2 cos^(-1)x`
so option (b) is true
`sin^(-1)sqrt(1-x)/(2)=sin^(-1)(sin(theta)/(2))=1/2theta=1/2cos^(-1)x`
`sin^(-1)sqrt(1-x)/(2)=sin^(-1)(sin(theta)/(2))=(theat)/(2)=(1)/(2)cosT^(-1)x`
so option (c ) is true