Question

Which of the following options is equal to $\sqrt{{\sec }^2\theta +cose{c}^2\theta }$ ?

• A. $\tan \theta +\cot \theta$
• B. $1$
• C. $cosec\theta +\sec \theta$
• D. $\text{cosec}\theta \sec \theta$

Answer 1

The correct option is D

$\text{cosec}\theta \sec \theta$

$\sqrt{{\text{sec}}^2\theta +{\text{cosec}}^2\theta }=\sqrt{1+{\text{tan}}^2\theta +1+{\text{cot}}^2\theta }$

$=\sqrt{{\text{tan}}^2\theta +2+{\text{cot}}^2\theta }$

$=\sqrt{{\text{tan}}^2\theta +2\left(\text{tan}\theta \right)\left(\text{cot}\theta \right)+{\text{cot}}^2\theta }\dots \dots (\text{tan}\theta \text{cot}\theta =1)$

$=\sqrt{(\text{tan}\theta +\text{cot}\theta {)}^2}$

$=\text{tan}\theta +\text{cot}\theta$

$=\frac{\text{sin}\theta }{\text{cos}\theta }+\frac{\text{cos}\theta }{\text{sin}\theta }$

$=\frac{{\text{sin}}^2\theta +{\text{cos}}^2\theta }{\text{sin}\theta \text{cos}\theta }$

$=\frac{1}{sin\theta cos\theta }$

$=\text{cosec}\theta \text{sec}\theta$