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

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

Prove that tan ^(-1) x+tan ^(-1) ((2 x)/(1-x^2))=tan ^(-1)((3 x-x^3)/(1-3 x^2)),|x|lt1/sqrt3

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

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

Prove that ""^(2n+1)P_(n-1)=((2n+1)!)/((n+2)!) and ""^(2n-1)P_n=((2n-1)!)/((n-1)!)

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 ((n+1)/(2))^(n) gt (n!)

cos ^(-1) ((3+5 cos x )/(5+3 cos x )) is equal to : a) tan ^(-1) ((1)/(2) tan ""(x)/(2)) b) 2 tan ^(-1) (2 tan ""(x)/(2)) c) (1)/(2) tan ^(-1) (2 tan ""(x)/(2)) d) 2 tan ^(-1) ((1)/(2) tan ""(x)/(2))

prove that tan^(-1) sqrt x = 1/2 cos^(-1)((1-x)/(1+x)), x in [0,1]

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