about to only mathematics

Options

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

Answer

C

Explanation

We have, `" " 0 lt x lt 1 `
Let `" " cot ^(-1) x = theta`

`rArr " " cot theta = x `
`rArr " " sin theta = (1)/(sqrt (1 + x ^(2))) = sin (cot ^(-1) x )`
and ` " " cot theta = ( x)/( sqrt (1 + x ^(2))) = cos ( cot ^(-1) x)`
Now, `sqrt ( 1 + x ^(2)) [{ x cos ( cot ^(-1) x ) + sin (cot ^(-1) x )}^(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) = x sqrt (1+ x ^(2))`