$3$\displaystyle{ \sin x=\frac{\hspace{2} 2t \hspace{2}}{\hspace{2} 1+ {t}^{2} \hspace{2}} }$

$3$\displaystyle{ \sin x=2\sin \frac{\hspace{2}x\hspace{2}}{\hspace{2}2\hspace{2}}\cdot \cos \frac{\hspace{2}x\hspace{2}}{\hspace{2}2\hspace{2}} }$
　　　　（2倍角の公式）⇒
　　　 $3$\displaystyle{ =2\cdot \frac{\hspace{2} \sin \frac{\hspace{2}x\hspace{2}}{\hspace{2}2\hspace{2}} \hspace{2}}{\hspace{2} \cos \frac{\hspace{2}x\hspace{2}}{\hspace{2}2\hspace{2}} \hspace{2}}\cdot {\cos }^{2}\frac{\hspace{2}x\hspace{2}}{\hspace{2}2\hspace{2}} }$
　　　 $3$\displaystyle{ =2\cdot \tan \frac{\hspace{2}x\hspace{2}}{\hspace{2}2\hspace{2}}\cdot \frac{\hspace{2}1\hspace{2}}{\hspace{2} 1+ {\tan }^{2}\frac{\hspace{2}x\hspace{2}}{\hspace{2}2\hspace{2}} \hspace{2}} }$
　　　 $3$\displaystyle{ =\frac{\hspace{2} 2\tan \frac{\hspace{2}x\hspace{2}}{\hspace{2}2\hspace{2}} \hspace{2}}{\hspace{2} 1+ {\tan }^{2}\frac{\hspace{2}x\hspace{2}}{\hspace{2}2\hspace{2}} \hspace{2}}=\frac{\hspace{2} 2t \hspace{2}}{\hspace{2} 1+ {t}^{2} \hspace{2}} }$