Prove that : `cos (2 sin^(-1) x) = 1-2x^2`

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

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

Prove that cos ^(3) x sin ^(2) x = (1)/(16) (2cos x - cos 3x - cos 5x).

Prove that 2cos^(-1)x=sin^(-1)(2xsqrt(1-x^2))

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

Prove that : cos^(-1) x = 2 cos^(-1) sqrt((1+x)/(2)) (ii) Prove that : tan^(-1)((cosx + sin x)/(cosx - sin x)) = (pi)/(4)+ x

Prove that : cos^(-1) x + cos^(-1) ((x)/(2) + (sqrt( 3-3x^2) )/( 2) ) = (pi)/ (3)

Prove that (1+cos x)/(sin x) = (cos( x/2))/(sin (x/2))

Prove that : cos 4x = 1 - 8 sin^2 x cos^2 x .

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