Question : If $\cot^2θ = 1 - e^2$, then the value of $\operatorname{cosec} θ + \cot^3θ \sec θ$ is:
Option 1: $\left(2-{e}^2\right)^ \frac{1}{2}$
Option 2: $\left(1-{e}^2\right)^ \frac{3}{2}$
Option 3: $\left(1-{e}^2\right)$
Option 4: $\left(2-{e}^2\right) ^\frac{3}{2}$
Correct Answer: $\left(2-{e}^2\right) ^\frac{3}{2}$
Solution :
Given,
$\cot^2θ = 1 - e^2$
Consider,
$\operatorname{cosec} θ + \cot^3θ \sec θ$
$=\frac{1}{\sinθ} + \frac{\cos^3θ}{\sin^3θ}\frac1{\cosθ}$
$=\frac{\sin^2θ+\cos^2θ}{\sin^3θ}$
$=\frac{1}{\sin^3θ}$ [As $\sin^2θ+\cos^2θ=1$]
$=\operatorname{cosec^3}θ$
Also, we know that,
$\operatorname{cosec^2}θ=1+\cot^2θ$
⇒ $\operatorname{cosec^2}θ=1+1-e^2$
⇒ $\operatorname{cosec^2}θ=2-e^2$
⇒ $\operatorname{cosec}θ=(2-e^2)^{\frac12}$
⇒ $\operatorname{cosec^3}θ=(2-e^2)^{\frac32}$
Hence, the correct answer is $(2-e^2)^{\frac32}$.
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