$If \sin \alpha = p , then the quadratic equation whose roots are \tan \frac{\alpha }{2} , cot\frac{\alpha }{2} is$

$\begin{array}{rcl}\sin \mathrm{\alpha }& =& \mathrm{P}\\ \tan \frac{\mathrm{\alpha }}{2}+\cot \frac{\mathrm{\alpha }}{2}& =& 2\mathrm{cosec} \mathrm{\alpha }=\frac{2}{\mathrm{P}}\\ \tan \frac{\mathrm{\alpha }}{2} \cot \frac{\mathrm{\alpha }}{2}& =& 1\end{array}$

$\tan \frac{\mathrm{\alpha }}{2},\cot \frac{\mathrm{\alpha }}{2}$  are roots of eqns

$\begin{array}{rcl}{\mathrm{x}}^2-\frac{2}{\mathrm{P}}\mathrm{x}+1& =& 0\\ & \Rightarrow & {\mathrm{Px}}^2=2\mathrm{Px}+\mathrm{P}=0\end{array}$