Question

Solve the trigonometric equation
$\frac{(\cot A-1+\text{cosec}A{)}^2}{(\cot A{)}^2-(1-\text{cosec}A{)}^2}$

Answer 1

$\frac{{(\cot A-1+\csc A)}^2}{{\cot }^{2}A-{(1-\csc A)}^2}$

$=\frac{{(\cot A-1+\csc A)}^2}{(\cot A-1+\csc A)(\cot A+1-\csc A)}$

$=\frac{(\cot A-1+\csc A)}{(\cot A-\csc A+1)}$

$=\frac{(\cot A+\csc A-1)}{(\cot A-\csc A+1)}$

$=\frac{(\cot A+\csc A-({\cot }^{2}A-{\csc }^{2}A))}{(\cot A-\csc A+1)}$

$=\frac{(\cot A+\csc A)(1-\csc A+\cot A)}{(\cot A-\csc A+1)}$

$=\cot A+\csc A$

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

$=\frac{2{\cos }^2\frac{A}{2}-1+1}{2\sin \frac{A}{2}\cos \frac{A}{2}}$

$=\frac{\cos \frac{A}{2}}{\sin \frac{A}{2}}$

$=\cot \frac{A}{2}$