Question

Prove that :
$\frac{\cot A+\cos ecA-1}{\cot A-\cos ecA+1}=\frac{1+\cos A}{\sin A}$

Answer 1

$\frac{cotA+cosecA-1}{cotA-cosA+1}=\frac{CotA+cosecA-1}{cotA-(cosecA-1)}$

$=\frac{co{t}^{2}A+cose{c}^{2}A+1-2cosecA+2cotAcosecA+2cotA}{co{t}^{2}A-cose{c}^{2}A-1+2cosecA}$

$=\frac{2cose{c}^{2}A-2cosecA+2cotA-cosecA+2cotA}{-2+2cosecA}$

$=\frac{2cose{c}^{2}A[cosec\theta -1]+2cotA[cosecA-1]}{(-1+cosec)}$

$=cosecA+cotA\Rightarrow \frac{1+cosA}{sinA}$