Question

Prove that ${\displaystyle \frac{\cot \text{A}+\text{cosec A}-1}{\cot \text{A}-\text{cosec A}+1}=\frac{1+\cos \text{A}}{\sin \text{A}}}$

Answer 1

LHS = ${\displaystyle \frac{\cot \text{A}+\text{cosec A}-1}{\cot \text{A}-\text{cosec A}+1}}$

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

= ${\displaystyle \frac{(\cot \text{A}+\text{cosec A})(\text{cosec A}+\cot \text{A})(\text{cosec A}-\cot \text{A})}{\cot \text{A}-\text{cosec A}+1}}$

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

= ${\displaystyle \frac{(\cot \text{A}+\text{cosec A})(1-\text{cosec A}+\cot \text{A})}{1-\text{cosec A}+\cot \text{A}}}$

= cot A + cosec A

= ${\displaystyle \frac{\cos \text{A}}{\sin \text{A}}+\frac{1}{\sin \text{A}}=\frac{\cos \text{A}+1}{\sin \text{A}}}$

= ${\displaystyle \frac{1+\cos \text{A}}{\sin \text{A}}}$

= RHS

Hence proved.