Question

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

• A. True
• B. False

Answer 1

The correct option is A True

Given

LHS

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

Dividing Numerator and Denominator by $\sin \theta$

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

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

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

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

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

$⟹\text{cosec}\theta +\cot \theta$