Question

Prove that $\frac{1+cosecA+\cot A}{1+cosecA-\cot A}=\frac{cosecA+\cot A-1}{\cot A-cosecA+1}$.

Answer 1

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

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

$(\sin A+\cos A+1)(\sin A+\cos A-1)=(1+\sin A-\cos A)(1+\cos A-\sin A)$

$(\sin A+\cos A{)}^2-1^2=1+\cos A-\sin A+\sin A+\sin A.\cos A-{\sin }^{2}A-\cos A-{\cos }^{2}A+\sin A.\cos A$

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

$1+2\sin A\cos A-1=1-({\sin }^{2}A+{\cos }^{2}A)+2\sin A.\cos A$

$2\sin A.\cos A=1-1+2\sin A.\cos A$

$\sin A.\cos A=\sin A.\cos A$ Hence Proved.