The value of `x` for which `"sin"(cot^(-1)(1+x))="cos"(tan^(-1)x)` is `1/2` (b) 1 (c) 0 (d) `-1/2`

Options

A. `(1)/(2)`
B. 1
C. 0
D. `-(1)/(2)`

Answer

D

Explanation

We have,
`"sin" {"cot"^(-1) (1+x)} = "sin" {"sin"^(-1)((1)/(sqrt(x^(2)+2x+2)))} = (1)/(sqrt(x^(2) + 2x +2))`
`"and, cos" ("tan"^(-1) x) = "cos" {"cos"^(-1)((1)/(sqrt(x^(2) + 1)))} = (1)/(sqrt(x^(2) + 1))`
`therefore "sin"{"cot"^(-1)(1+x)} = "cos"("tan"^(-1)x)`
`rArr (1)/(sqrt(x^(2) + 2x +2)) = (1)/(sqrt(x^(2) + 1))`
`rArr x^(2) + 2x +2 = x^(2) + 1 rArr 2x + 1 = 0 rArr x = -(1)/(2)`