Prove that : ` cos [ tan^(-1) { sin (cot^(-1) x)}]= sqrt((x^2 +1)/(x^2 +2))`

Explanation

`cot^-1(x) = sin^-1(1/(sqrt(1+x^2)))` if `x gt 0`

`cot^-1(x) = sin^-1(pi-1/(sqrt(1+x^2)))` if `x lt 0`

In both cases,

`sin(cot^-1x) = 1/(sqrt(1+x^2)`

`=>tan^-1(sin(cot^-1x)) = tan^-1(1/(sqrt(1+x^2))) = cos^-1(sqrt(1+x^2)/sqrt(2+x^2))`

`:. costan^-1(sin(cot^-1x)) = coscos^-1(sqrt(1+x^2)/sqrt(2+x^2))`

`=>costan^-1(sin(cot^-1x)) = (sqrt(x^2+1)/sqrt(x^2+2)).`