For given hyperbola the eccentricity is given as
e2=1+cos2θsin2θ=1+tan2θ=sec2θ
⇒e=secθ
∴ Length of latus rectum l=cosθ2sin2θ=secθ2tan2θ
⇒l=e2(e2−1)=2(e−e1)
On differentiating w.r.t. e, we get
dedl=2(1+e21)>0
∴ l is an increasing function
∴lmin=2(2−21)=3
∴ Range of latus rectum is (3,∞).