Question

Prove that: $\frac{\sin A}{(\cot A+cosecA)}=2+\frac{\sin A}{(\cot A-cosecA)}$

Answer 1

$\frac{\sin A}{\cot A+\csc A}=2+\frac{\sin A}{\cot A-\csc A}\frac{\sin A}{\cot A+\csc A}-\frac{\sin A}{\cot A-\csc A}=2LHS=\frac{\sin A}{\frac{\cos A}{\sin A}+\frac{1}{\sin A}}-\frac{\sin A}{\frac{\cos A}{\sin A}-\frac{1}{\sin A}}=\frac{{\sin }^{2}A}{\cos A+1}-\frac{{\sin }^{2}A}{\cos A-1}={\sin }^{2}A\left(\frac{\cos A-1-\cos A-1}{{\cos }^{2}A-1}\right)=\frac{-2{\sin }^{2}A}{-{\sin }^{2}A}=2=RHS$