Question

Solve:

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

• A. $\frac{1+\cos \theta }{\sin \theta }$
• B. $\frac{\sin \theta }{1-\cos \theta }$
• C. $\frac{1-\cos \theta }{\sin \theta }$
• D. $\frac{\cos \theta }{1-\sin \theta }$

Answer 1

The correct option is A $\frac{1+\cos \theta }{\sin \theta }$

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

$=\frac{\cot \theta +\text{cosec}\theta -({\text{cosec}}^2\theta -{\cot }^2\theta )}{\cot \theta -\text{cosec}\theta +1}$

$=\frac{(\cot \theta +\text{cosec}\theta )[1+\cot \theta -\text{cosec}\theta ]}{1+\cot \theta -\text{cosec}\theta }$

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

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

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