Question

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

Answer 1

$\begin{array}{c}wehave\\ \cos ecA-\cot A=\frac{1-\cos A}{1+\cos A}\\ Now\\ L.H.S=\left(\cos ecA-\cot A\right)\\ =\left(\frac{1}{\sin A}-\frac{\cos A}{\sin A}\right)\\ ={\left(\frac{1-\cos A}{\sin A}\right)}^2\\ =\frac{{\left(1-\cos A\right)}^2}{{\sin }^{2}A}\\ =\frac{\left(1-\cos A\right)\left(1-\cos A\right)}{1-{\cos }^{2}A}\\ =\frac{\left(1-\cos A\right)\left(1-\cos A\right)}{\left(1+\cos A\right)\left(1-\cos A\right)}\\ =\frac{1-\cos A}{1+\cos A}=R.H.S\\ Hence,L.H.S=R.H.Sproved.\end{array}$