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

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

Solve the following equations: cos(tan^-1x = sin(cot^-1 (x+1)

Prove that following : cos^-1 x = 2 sin^-1 sqrt(1-x)/(2) = 2cos^-1 sqrt(1+x)/(2), |x| le 1

Prove the following: cos 4x = 1 - 8sin^2 x cos^2 x

Prove that following : 2 sin^-1 x = cos^-1 (1-2x^2), 0 le x le 1

Prove that : 1/2 tan^-1x = cos^-1{(1+sqrt(1+x^2))/(2sqrt(1+x^2))}^(1/2)

Differentiate the following w.r.t. as indicated: tan^-1(x/(1+sqrt(1-x^2))) w.r.t. sin (2 cot^-1sqrt((1+x)/(1-x))

Solve the following equations. tan^-1 (sqrt(x^2 + x) + sin^-1 sqrt(x^2 + x + 1) = pi/2

Solve the following equations tan^(-1)""(2x)/(1-x^(2))+cot^(-1)""(1-x^(2))/(2x)=(pi)/(3)

Prove that sec [cot^-1{sin(tan^-1 (cosec(cos^-1 a)))}] = sqrt(3-a^2), where |a| < 1