" 19."quad tan^(-1)[(x-sqrt(a^(2)-x^(2)))/(x+sqrt(a^(2)-x^(2)))]

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

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

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

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

Show that : tan^(-1)[(sqrt(1+x^(2))+sqrt(1-x^(2)))/(sqrt(1+x^(2))-sqrt(1-x^(2)))]=pi/4+1/2cos^(-1)x^(2) .

If y=tan^(-1)[(x-sqrt(1-x^(2)))/(x+sqrt(1-x^(2)))]," then "(dy)/(dx)=

tan^(-1)[(sqrt(1+x^(2))+sqrt(1-x^(2)))/(sqrt(1+x^(2))-sqrt(1-x^(2)))],|x|<(1)/(2),x!=0

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) .

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)