$sec \theta + tan^3 \theta cosec \theta = sec \theta + tan^2 \theta \frac{sin \theta}{cos \theta} . \frac{1}{sin \theta}$
$= sec \theta + tan^2 \theta \ sec\theta$
$= sec \theta \left(1 + tan^2 \theta\right)$
$= sec \theta \left(sec^2 \theta\right)$
$= sec^3 \theta$
$= \left(sec^2 \theta\right)^{\frac{3}{2}}$
$= \left(1 + tan^2 \theta\right)^{\frac{3}{2}}$
$= \left(1 + 1 - e^2 \right)^{\frac{3}{2}}$
$= \left(2 - e^2 \right)^{\frac{3}{2}}$
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