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)`

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)

If the chord through the points (a sec theta, b tan theta) and (a sec phi, b tan phi) on the hyperbola x^2/a^@ - y^2/b^2 = 1 passes through a focus, prove that tan theta/2 tan phi/2 + (e-1)/(e+1) = 0 .

If alpha and beta are eccentric angles of the ends of a focal chord of the ellipse x^2/a^2 + y^2/b^2 =1 , then tan alpha/2 .tan beta/2 is (A) (1-e)/(1+e) (B) (e+1)/(e-1) (C) (e-1)/(e+1) (D) none of these

If (a sec theta,b tan theta) and (a sec phi,b tan phi) be two coordinate of the ends of a focal chord passing through (ae,0) of (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 then tan((theta)/(2))tan((phi)/(2)) equals to

If the chord joining the points (a sec theta_(1),b tan theta_(1)) and (a sec theta_(2),b tan theta_(2)) on the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 is a focal chord,then prove that tan((theta_(1))/(2))tan((theta_(2))/(2))+(ke-1)/(ke+1)=0, where k=+-1

If x+y=45^(@) then prove that :(1+tan x)(1+tan y)=2(cot x-1)(cot y-1)=2

IF alpha , beta are eccentric angles of end points of a focal chord of the ellipse x^(2)/a^(2) + y^(2)/b^(2) =1 then tan(alpha /2) tan (beta/2) is equal to

if y=tan^(-1)((2x)/(1-x^(2))) then prove that (dy)/(dx)=(2)/(1+x^(2))