Question

Prove that $\frac{(cotA-\cos A)}{(cotA+\cos A)}=\frac{(\cos ecA-1)}{(\cos ecA+1)}$

Answer 1

We have to prove that $\frac{(cotA-\cos A)}{(cotA+\cos A)}=\frac{(\cos ecA-1)}{(\cos ecA+1)}$

$L.H.S=\frac{(cotA-\cos A)}{(cotA+\cos A)}$

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

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

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

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

$=R.H.S.$

Hence, it is proved that $\frac{(cotA-\cos A)}{(cotA+\cos A)}=\frac{(\cos ecA-1)}{(\cos ecA+1)}$.