if `tan ^-1 (( sqrt (1+x^2)-1)/x) = pi/45` then:

Prove that tan^-1((sqrt1+X^2) + 1)/x) = pi/2 - 1/2 tan^-1 x .

Consider y=tan^-1((sqrt(1+x^2)+1)/x) . Put x= tan theta and show that (sqrt(1+x^2)+1)/x=tan(pi/2-theta/2) .

Prove that tan^(-1).(1)/(sqrt(x^(2) -1)) = (pi)/(2) - sec^(-1) x, x gt 1

Prove that tan^(-1)((1)/(sqrt(x^(2) -1))) = (pi)/(2) - sec^(-1) x, x gt 1

Prove that tan^(-1).(1)/(sqrt(x^(2) -1)) = (pi)/(2) - sec^(-1) x, x gt 1

Prove that tan^(-1).(1)/(sqrt(x^(2) -1)) = (pi)/(2) - sec^(-1) x, x gt 1

Prove that tan^(-1) ((1-sqrt(x))/(1+sqrt(x))) = pi/4 - tan^(-1) sqrt(x) , "where" x gt 0

If tan^(-1) sqrt((1-sqrt(x))/(1+sqrt(x))) = (pi)/(4)-(alpha)/(2) , then express tan^(2)alpha in terms of x.

If tan^(-1)(1+x)+tan^(-1)(1-x)=(pi)/(6), then prove that x^(2)=2sqrt(3).

Number of values of x satisfying simultaneously sin^(-1) x = 2 tan^(-1) x " and " tan^(-1) sqrt(x(x-1)) + cosec^(-1) sqrt(1 + x - x^(2)) = pi/2 , is