万能公式

$\sin \theta =2\sin {\displaystyle \frac{\theta }{2}}\cos {\displaystyle \frac{\theta }{2}}={\displaystyle \frac{2\sin \frac{\theta }{2}\cos \frac{\theta }{2}}{{\sin }^2\frac{\theta }{2}+{\cos }^2\frac{\theta }{2}}}={\displaystyle \frac{2\tan \frac{\theta }{2}}{1+{\tan }^2\frac{\theta }{2}}}$
$\cos \theta ={\cos }^2{\displaystyle \frac{\theta }{2}}-{\sin }^2{\displaystyle \frac{\theta }{2}}={\displaystyle \frac{{\cos }^2\frac{\theta }{2}-{\sin }^{\frac{\theta }{2}}}{{\sin }^2\frac{\theta }{2}+{\cos }^2\frac{\theta }{2}}}={\displaystyle \frac{1-{\tan }^2\frac{\theta }{2}}{1+{\tan }^2\frac{\theta }{2}}}$
$\tan \theta ={\displaystyle \frac{2\tan \frac{\theta }{2}}{1-{\tan }^2\frac{\theta }{2}}}$

$y={\displaystyle \frac{x}{1+x^2}}\mathrm{的}\mathrm{值}\mathrm{域}$
$\mathrm{设}x=\tan \theta ,\mathrm{则}$
$y={\displaystyle \frac{x}{1+x^2}}\Rightarrow y={\displaystyle \frac{1}{2}}{\displaystyle \frac{2\tan \theta }{1+{\tan }^2\theta }}={\displaystyle \frac{1}{2}}\sin 2\theta$
$\mathrm{所}\mathrm{以}\mathrm{值}\mathrm{域}\mathrm{为}[-{\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}]$