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

Prove that cos (tan^(-1) (sin (cot^(-1) x))) = sqrt((x^(2) + 1)/(x^(2) + 2))

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

Prove the following: "cos"{tan^(-1){sin(cot^(-1)x)}}= sqrt((1+x^2)/(2+x^2))

Prove that cos ^(-1) x = 2 sin ^(-1).sqrt(1-x)/(2)

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

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

Prove that tan^(-1)[(sqrt(1+x^2)+sqrt(1-x^2))/(sqrt(1+x^2)-sqrt(1-x^2))]=pi/4+1/2cos^(-1)x^2

If x lt 0 , then prove that cos^(-1) x = pi - sin^(-1) sqrt(1 - x^(2))

The expression 1/(sqrt(2)){(sincot^(- 1)costan^(- 1)t)/(costan^(- 1)sincot^(- 1)sqrt(2)t)}*{sqrt((1+2t^2)/(2+t^2))} can take the value

Prove that tan^(-1).(1)/(sqrt(x^(2) -1)) = (pi)/(2) - sec^(-1) x, x gt 1