If cot² θ = 3 - a², then cosec θ + cot³ θ. sec θ =?

(4 - a²)^{\(\frac{3}{2}\)} 

Let put θ = 45°

⇒ cot² θ = 3 - a²

⇒ 1 = 3 -  a²

⇒ a² = 2

and

⇒ cosec θ + cot³θ . sec θ = cosec 45° + cot³ 45° × sec 45°

= \(\sqrt {2}\) + \(\sqrt {2}\) = 2\(\sqrt {2}\)

Now put a² = 2 in options and find answer

(4 - a²)^{\(\frac{3}{2}\)} = (2)^{\(\frac{3}{2}\)} = 2\(\sqrt {2}\) 

Hence, option (1) is correct answer.