Question

Prove the following,
(i) $\sqrt{\frac{1-\cos A}{1+\cos A}}=\cos ecA-cotA$

Answer 1

$$\begin{gathered} \mathrm{LHS}=\sqrt{\frac{1-\cos A}{1+\cos A}} \\ =\sqrt{\frac{\left(1-\cos A\right)\left(1-\cos A\right)}{\left(1+\cos A\right)\left(1-\cos A\right)}} \\ =\sqrt{\frac{{\left(1-\cos A\right)}^2}{1-{\cos }^{2}A}} \\ =\sqrt{\frac{{\left(1-\cos A\right)}^2}{{\sin }^{2}A}} \\ =\frac{1-\cos A}{\sin A} \end{gathered}$$

$$\begin{gathered} =\frac{1}{\sin A}-\frac{\cos A}{\sin A} \\ =\cos ecA-cotA \\ =\mathrm{RHS} \\ \mathrm{Hence},\mathrm{proved}. \end{gathered}$$