If `x lt 0`, then prove that `cos^(-1)x=pi+tan^(-1)""(sqrt(1-x^2))/x`

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

Prove that tan^(-1)((sqrt(1+x^2)-1)/x)=1/2 tan^(-1)x .

Prove that cosec^-1x+sec^-1x=pi/2

prove that 3 cos^(-1) x = cos^(-1) (4x^3 - 3x), x in [1/2,1]

Prove that cos^(-1)((1-x^(2n))/(1+x^(2n)))=2 tan^(-1)x^n

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

If x takes negative permissible values , then sin^(-1) x= a) cos^(-1)sqrt(1-x^2) b) -cos^(-1)sqrt(1-x^2) c) cos^(-1)sqrt(x^2-1) d) pi-cos^(-1)sqrt(1-x^2)

Prove that cos^(-1){sqrt((1+x)/2)}=(cos^(-1)x)/2,-1ltxlt1

If x in (0,1) then the value of tan^(-1)((1-x^2)/(2x))+cos^(-1)((1-x^2)/(1+x^2)) is equal to a) -pi/2 b)0 c) pi/2 d) pi