Question

Which of the following is the correct expression for $\frac{\text{cos}\theta }{\text{cosec}\theta +1}+\frac{\text{cos}\theta }{\text{cosec}\theta -1}$?

• A. $2\text{tan}\theta$
• B. $\text{tan}\theta$
• C. $0$
• D. $\text{-tan}\theta$

Answer 1

The correct option is A $2\text{tan}\theta$

$\frac{\text{cos}\theta }{\text{cosec}\theta +1}+\frac{\text{cos}\theta }{\text{cosec}\theta -1}$

$=\text{cos}\theta [\frac{1}{\text{cosec}\theta +1}+\frac{1}{\text{cosec}\theta -1}]$

$=\text{cos}\theta \left[\frac{\text{cosec}\theta -1+\text{cosec}\theta +1}{{\text{cosec}}^2\theta -1}\right]$

$=\frac{\text{cos}\theta \left(2\text{cosec}\theta \right)}{{\text{cot}}^2\theta }(\because 1+co{t}^2\theta =cose{c}^2\theta )$

$=2\frac{\text{cos}\theta \times si{n}^2\theta }{\text{sin}\theta \times {\text{cos}}^2\theta }$

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

$=2\text{tan}\theta$