Question

If $\cos \theta +cosec\theta =2$
then the value of $\frac{1+\cos \theta }{1-\cos \theta }$ is :

• A. $2$
• B. $4$
• C. $\frac{1}{2}$
• D. $\frac{1}{4}$

Answer 1

Answer: (B)

$4$

Given $\cot \theta +cosec\theta =2$

Then

Multiply the given eq. by $(cosec\theta -\cot \theta )$

$\Rightarrow (cosec\theta +\cot \theta )(cosec\theta -\cot \theta )=2(cosec\theta -\cot \theta )$

$\Rightarrow 1=2(cosec\theta -\cot \theta )$

$[\because cose{c}^2\theta -{\cot }^2\theta =1]$

$\Rightarrow \frac{1}{2}=(cosec\theta -\cot \theta )$

and $2=cosec\theta +\cot \theta$ (Given)

$\Rightarrow 2cosec\theta =\frac{5}{2}$

$\Rightarrow cosec\theta =\frac{5}{4}$

$\Rightarrow \sin \theta =\frac{4}{5}[\because \sin \theta =\frac{1}{cosec\theta }]$

and $\cos \theta =\sqrt{1-{\sin }^2\theta }$

$=\sqrt{1-{\left(\frac{4}{5}\right)}^2}=\sqrt{1-\frac{16}{25}}$

$=\sqrt{\frac{9}{25}}=\frac{3}{5}$

Then

$\frac{1+\cos \theta }{1-\cos \theta }=\frac{1+\frac{3}{5}}{1-\frac{3}{5}}=\frac{8}{5}\times \frac{5}{2}=4$