`If"sin"{cot^(-1)(x+1)}="cos"(tan^(-1)x),` then find `xdot`

Options

A. `(1)/(3)`
B. `(1)/(2)`
C. `-(1)/(2)`
D. `-(1)/(3)`

Answer

C

Explanation

Given , `sin[cot^(-1)(x+1)]=cos(tan^(-1)x)` . . . (i)
Let `cot^(-1)(x+1)=theta`
`rArrsintheta=(1)/(sqrt(x^(2)+2x+2))`
`rArrtheta=sin^(-1)((1)/(sqrt(x^(2)+2x+2)))`
`cot^(-1)(x+1)=sin^(-1)((1)/(sqrt(x^(2)+2x+2)))`

Let `tan^(-1)x=phirArrtanvarphi=x`
`rArrcosvarphi=(1)/(sqrt(x^(2)+1))`
`rArrvarphi=cos^(-1)((1)/(sqrt(x^(2)+1)))`
`rArrtan^(-1)x=cos^(-1)((1)/(sqrt(x^(2)+1)))`
Now , from Eq . (i) we get
`sin[sin^(-1)((1)/(sqrt(x^(2)+2x+2)))]=cos[cos^(-1)((1)/(sqrt(1+x^(2))))]`
`rArr(1)/(sqrt(x^(2)+2x+2))=(1)/(sqrt(1+x^(2)))`
`rArrsqrt(x^(2)+2x+2)=sqrt(1+x^(2))`
`rArrx^(2)+2x+2=1+x^(2)`
`rArr2x+2=1`
`rArr2x=-1`
`rArrx=-(1)/(2)`