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

Prove that : sin(tan^(-1)1) = 1/sqrt2

Prove that : tan^-1[(sqrt(1+x^2) - sqrt(1-x^2))/(sqrt1+x^2 + sqrt(1-x^2))] = pi/4 - 1/2cos^-1x^2

Prove that tan^-1((sqrt1+X^2) + 1)/x) = pi/2 - 1/2 tan^-1 x .

Two concentric ellipses are such that the foci of one are on the other and their major axes are equal. Let ea n de ' be their eccentricities. Then. the quadrilateral formed by joining the foci of the two ellipses is a parallelogram the angle theta between their axes is given by theta=cos^(-1)sqrt(1/(e^2)+1/(e^('2))=1/(e^2e^('2))) If e^2+e^('2)=1, then the angle between the axes of the two ellipses is 90^0 none of these

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

Evaluate: int(e^tan^((-1)x))/((1+x^2))[((sec^(-1)sqrt(1+x^2))^2+cos^(-1)((1-x^2)/(1+x^2))]dx(x >0)dot

If e^x + e^y = e^(x +y) , prove that (dy)/(dx) = (e^x (e^y - 1))/(e^y (e^x - 1))

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

If xlog_e(log_ex)-x^2+y^2=4 then ((dy)/(dx))_(atx=e) is equal to (A) (2e+1)/ sqrt(4+e^2) (B) e/(2sqrt(4+e^2)) (C) (2e+1)/(2(4+e^2)) (D) (2e-1)/(2sqrt(4+e^2))