8.(1)/(2)tan^(-1)x=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)))).

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

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

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

tan[2Tan^(-1)((sqrt(1+x^(2))-1)/x)]=

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