Question

If ${\tan }^2\theta =1-e^2,$ then the value of $\sec \theta +{\tan }^3\theta \csc \theta$ is ?

• A. ${\left(2-e^2\right)}^{\frac{3}{2}}$
• B. ${\left(2-e^2\right)}^{\frac{1}{2}}$
• C. ${\left(2+e^2\right)}^{\frac{1}{2}}$
• D. ${\left(2+e^2\right)}^{\frac{3}{2}}$

Answer 1

Answer: (A)

${\left(2-e^2\right)}^{\frac{3}{2}}$

${\tan }^2\theta =1-e^2$

$e^2=1-{\tan }^2\theta$

Also ${\sec }^2\theta =2-e^2$

The above expression can be expressed in terms of powers of $\sec \theta$

$\sec \theta +{\tan }^3\theta \csc \theta =(2-(1-{\tan }^2\theta ){)}^n$

$\frac{1}{\cos \theta }+\frac{{\sin }^3\theta }{{\cos }^3\theta }\times \frac{1}{\sin \theta }=(1+{\tan }^2\theta {)}^n$

$\frac{1}{{\cos }^3\theta }=({\sec }^2\theta {)}^n$

$3=2n$

$n=\frac{3}{2}$

$\Rightarrow \sec \theta +{\tan }^3\theta \csc \theta =(2-(1-{\tan }^2\theta ){)}^{\frac{3}{2}}$

$\Rightarrow \sec \theta +{\tan }^3\theta \csc \theta =(2-e^2{)}^{\frac{3}{2}}$