(d)/(8x^(***))(1)/(2)tan^(-1)x=cos^(-1){(1+sqrt(1+x^(2)))/(2sqrt(1+x^(2)))}^((1)/(2))

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

show that , (1) /(2) tan ^(-1) x = cos^(-1) sqrt((1+sqrt(1+x^(2)))/(2sqrt(1+x^(2)))).

tan^(-1)((sqrt(1+x^(2))+sqrt(1-x^(2)))/(sqrt(1+x^(2))-sqrt(1-x^(2))))

tan^(-1)((sqrt(1+x^(2))+sqrt(1-x^(2)))/(sqrt(1+x^(2))-sqrt(1-x^(2))))

(d)/(dx)[cos^(-1)(x sqrt(x)-sqrt((1-x)(1-x^(2))))]=(1)/(sqrt(1-x^(2)))-(1)/(2sqrt(x-x^(2)))(-1)/(sqrt(1-x^(2)))-(1)/(2sqrt(x-x^(2)))(1)/(sqrt(1-x^(2)))+(1)/(2sqrt(x-x^(2)))(1)/(sqrt(1-x^(2)))0 b.1/4c.-1/4d none of these

d//dx[tan^(-1)((sqrt(x^(2)+a^(2))+x)/(sqrt(x^(2)+a^(2))-x))^(1//2)]

Derivative of tan ^(-1) ((sqrt( 1+x^(2))-1)/( x)) w.r.cos ^(-1) sqrt((1+sqrt( 1+x^(2)))/( 2sqrt(1+x^(2)))) is

y=tan^(-1)((sqrt(1+x^2)+sqrt(1-x^2))/(sqrt(1+x^2)-sqrt(1-x^2)))