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

Prove that costan^(-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) (cot^(-1)x)}]=sqrt((x^2+1)/(x^2+2))

If 1/(sqrt(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

prove that cos^(-1)x=2sin^(-1)sqrt((1-x)/(2))=2cos^(-1)sqrt((1+x)/(2))

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

Prove that : tan^-1[(sqrt(1+x^2) - sqrt(1-x^2))/(sqrt1+x^2 + sqrt(1-x^2))] = pi/4 - 1/2cos^-1x^2

Prove the followings : cos^(-1)x=2sin^(-1)sqrt((1-x)/2)=2cos^(-1)sqrt((1+x)/2)