Question

$cosecA(1+cosA)(cosecA-cotA)=1$

Answer 1

$\csc A(1+\cos A)(\csc A-\cot A)=1$

$LHS:\frac{1}{\sin A}[1+\cos A][\frac{1}{\sin A}-\frac{\cos A}{\sin A}][\because \csc A=\frac{1}{\sin A},\cot A=\frac{\cos \overrightarrow{A}}{\sin \overrightarrow{A}}]$

$=\frac{A}{\sin A}(1+\cos A)\frac{(1-\cos A)}{\sin A}$

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

$=\frac{{\sin }^{2}A}{{\sin }^{2}A}=1=RHS$