Question

Evaluate: $\frac{\cos \theta -\sin \theta +1}{\cos \theta +\sin \theta -1}$.

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

Answer 1

Answer: (A)

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

Given that $\frac{\cos \theta -\sin \theta +1}{\cos \theta +\sin \theta -1}$

divide both Numerator and Denominator by $\sin \theta$

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

=$\frac{\cot \theta +\text{coesc}\theta -1}{\cot \theta +1-\text{coesc}\theta }$

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

=$\frac{\cot \theta +\text{coesc}\theta (1-(\text{coesc}\theta -\cot \theta )}{\cot \theta -\text{coesc}\theta +1}$

=$\text{coesc}\theta +\cot \theta$.