If $tan^2θ = 1-a^2$, then the value of $secθ+ tan^3θ cosecθ$ is:

$(2-a^2)^{\frac{3}{2}}$

$secθ+ tan^3θ cosecθ$

= \(\frac{1}{cosθ}\) + \(\frac{sin³θ}{cos³θ}\) × \(\frac{1}{sinθ}\)

= \(\frac{1}{cosθ}\)  [ 1 + \(\frac{sin²θ}{cos²θ}\) ]

= \(\frac{1}{cosθ}\)  [  \(\frac{ cos²θ + sin²θ }{cos²θ}\) ] 

= \(\frac{1}{cosθ}\) × \(\frac{ 1 }{cos²θ}\)

= \(\frac{ 1 }{cos³θ}\)

= sec³θ

As ,tan²θ = 1 - a²

sec²θ - 1 = 1 - a²      

( tan²θ  = sec²θ - 1 )

sec²θ = 2 - a²    

sec³θ = $(2-a^2)^{\frac{3}{2}}$