Question

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

Answer 1

$\frac{\cot A-\cos A}{\cot A+\cos A}$

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

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

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

$=\frac{\cos ecA-1}{\cos ecA+1}$