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

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

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

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

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

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

e_(1) and e_(2) are respectively the eccentricites of a hyperbola and its conjugate. Prove that (1)/(e_1^(2))+(1)/(e_2^2) =1

e_(1) and e_(2) are eccentricities of hyperbola and conjugate hyperbola respectively then prove that (1)/(e_(1)^(2))+(1)/(e_(2)^(2))=1.

If e_1 and e_2 are the eccentricities of a hyperbola and its conjugate then prove that (1)/(e_1^2)+(1)/(e_2^2)=1 .

If e_1, e_2 , are the eccentricities of a hyperbola, its conjugate hyperbola, prove that (1)/(e_1^2)+(1)/(e_2^2)=1 .

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