Prove that `costan^(-1)sincot^(-1)x=sqrt((x^2+1)/(x^2+2))`

1.	Prove that `costan^(-1)sincot^(-1)x=sqrt((x^2+1)/(x^2+2))`
Answer» `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)).`

Discussion

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