Prove that `(tan^(-1)1/e)^2+(2e)/((e^2+1)<(tan^(-1)e)^2+2/(sqrt(e^2+1))`

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

If e and e' the eccentricities of a hyperbola and its conjugate,prove that (1)/(e^(2))+(1)/(e^(2))=1

Prove that int e^x (tan^-1 x+1/(1+x^2)) dx=e^x tan^-1x+c

Prove that 2tan^(-1)(cos e ctan^(-1)x-tancot^(-1)x)=tan^(-1)x(x!=0)dot

If the eccentricity of the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1and(y^(2))/(b^(2))-(x^(2))/(a^(2))=1" are "e_(1)ande_(2) respectively then prove that : (1)/(e_(1)^(2))+(1)/(e_(2)^(2))

If e_(1) and e_(2) are the eccentricities of the hyperbola and its conjugate hyperbola respectively then (1)/(e_(1)^(2))+(1)/(e_(2)^(2)) is equal to

int e^(tan^(-1)x)((1)/(1+x^(2)))dx=

If (a sec theta;b tan theta) and (a sec phi;b tan phi) are the ends of the focal chord of (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 then prove that tan((x)/(a))tan((phi)/(2))=(1-e)/(1+e)