$3$\displaystyle{{\sin }^{2} $ \frac{\hspace{2}{\alpha}\hspace{2}}{\hspace{2}2\hspace{2}} $ =\frac{\hspace{2} 1- \cos {\alpha} \hspace{2}}{\hspace{2}2\hspace{2}}}$

2倍角の公式より

$3$\displaystyle{\begin{array}{lll} \cos \alpha & =\cos $ 2\cdot \frac{\hspace{2}\alpha \hspace{2}}{\hspace{2}2\hspace{2}} $ & \vspace{6}\\ & ={ \cos }^{2}\frac{\hspace{2}\alpha \hspace{2}}{\hspace{2}2\hspace{2}}- { \sin }^{2}\frac{\hspace{2}\alpha \hspace{2}}{\hspace{2}2\hspace{2}} & \vspace{6}\\ & = \{ 1- { \sin }^{2} $ \frac{\hspace{2}\alpha \hspace{2}}{\hspace{2}2\hspace{2}} $ \} - { \sin }^{2} $ \frac{\hspace{2}\alpha \hspace{2}}{\hspace{2}2\hspace{2}} $ & \vspace{6}\\ & =1- { \sin }^{2} $ \frac{\hspace{2}\alpha \hspace{2}}{\hspace{2}2\hspace{2}} $ & \vspace{6}\\ \end{array}}$

よって

　 $3$\displaystyle{{\sin }^{2} $ \frac{\hspace{2}{\alpha}\hspace{2}}{\hspace{2}2\hspace{2}} $ =\frac{\hspace{2} 1- \cos {\alpha} \hspace{2}}{\hspace{2}2\hspace{2}}}$ 　

⇒半角の公式