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

If (asectheta;btantheta) and (asecphi; btanphi) 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)

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

If e^(cot^(-1)x) Prove that (1+x^(2))y_(2) +(2x + 1)y_(1) = 0 .

Prove that (d^n)/(dx^n)(e^(2x)+e^(-2x))=2^n[e^(2x)+(-1)^n e^(-2x)]

Prove that: cos e c(tan^(-1)("cos"(cot^(-1)("sec"(sin^(-1)a)))))=sqrt(3-a^2), where a in [0,1]

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

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

integrate (e^(2x))/(e^(2x)+1)

Show that int_(0)^(1)(e^(x))/(1+e^(2x))dx=tan^(-1)(e)-pi/(4)

The value of int_1^e((tan^(-1)x)/x+(logx)/(1+x^2))dxi s tane (b) tan^(-1)e tan^(-1)(1/e) (d) none of these