Question

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

Answer 1

$\frac{\cot A+cosecA-1}{\cot A-cosecA+1}=\frac{1+\cos A}{\sin A}=\frac{\cot A+cosecA-(cose{c}^{2}A-{\cot }^{2}A)}{\cot A-cosecA+1}[\because cose{c}^{2}A-{\cot }^{2}A=1]=\frac{(\cot A+cosecA)-(\cot A+cosecA)(cosecA-\cot A)}{\cot A-cosecA+1}=\frac{(\cot A+cosecA)(1-cosecA+\cot A)}{\cot A-cosecA+1}=\frac{\cos A}{\sin A}=\frac{1}{\sin A}=\frac{\cos A+1}{\sin A}=\frac{1+\cos A}{\sin A}=RHS$