<s>[INST] How to prove that $(1+\cos a)/(\sin a)=(\sin a)/(1-\cos a)$? How can I prove this relation $(1+\cos a)/(\sin a)=(\sin a)/(1-\cos a)$ ?
I tried to start from relation $\cos^2a+\sin^2a=1$ but relation went crazy with lot of $\cos$ and $\sin$ and $\sin^2$.  
 [/INST] $$\begin{align}\dfrac{1+\cos a}{\sin a} & = \dfrac{(1+\cos a)\cdot(1-\cos a)}{\sin a(1-\cos a)}\\ \, \\
&=\dfrac{1-\cos^2a}{\sin a(1-\cos a)}\quad\text{since: $(a+b)\cdot(a-b)=a^2-b^2$}\\\,\\
&=\dfrac{\cos^2a+\sin^2a-\cos^2a}{\sin a(1-\cos a)}\quad\text{since: $\cos^2a+\sin^2a=1$}\\\,\\
&=\dfrac{\sin^2a}{\sin a(1-\cos a)}\\\,\\&=\dfrac{\sin a\cdot\sin a}{\sin a(1-\cos a)}\\\,\\
&=\boxed{\dfrac{\sin a}{1-\cos a}}
\end{align}$$
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