If `0ltxlt1` then
`sqrt(1+x^(2))[{x cos )(cot^(-1)x)+sin(cot^(-1)x}^(2)-1]^(-//2)`

Options

A. `(x)/(sqrt(1+x^(2)))`
B. x
C. `xsqrt(1+x^(2))`
D. `sqrt(1+x^(2))`

Answer

C

Explanation

We have , `0ltxlt1`
Now , `sqrt(1+x^(2))[{xcos(cot^(-1)x)+sin(cot^(-1)x)}^(2)-1]^(1//2)`
`=sqrt(1+x^(2))[{xcos("cos"^(-1)(x)/(sqrt(1+x^(2))))+sin("sin"^(-1)(1)/(sqrt(1+x^(2))))}^(2)-1]^(-1//2)`
`=sqrt(1+x^(2))[{x(x)/(sqrt(1+x^(2)))+(1)/(sqrt(1+x^(2)))}^(2)-1]^(1//2)`
`=sqrt(1+x^(2))[((1+x^(2))/(sqrt(1+x^(2))))^(2)-1]^(1//2)`
`=sqrt(1+x^(2))[1+x^(2)-1]^(1//2)`
`=xsqrt(1+x^(2))`