背景：

        最近刷题发现有些中学的基础没打扎实，导致看到题虽然懂大概，但是最关键的也是最重要的步骤还是没理解，于是重新整理一下中学阶段该记下但是忘却的一些重要公式。供日后查阅。

• 倍角公式：

  $\begin{array}{rl}& \sin 2\alpha =2\sin \alpha \cos \alpha ,\cos 2\alpha ={\cos }^2\alpha –{\sin }^2\alpha =1-2{\sin }^2\alpha =2{\cos }^2\alpha -1\\ & \sin 3\alpha =-4{\sin }^3\alpha +3\sin \alpha ,\cos 3\alpha =4{\cos }^3\alpha –3\cos \alpha \\ & {\sin }^2\alpha =\frac{1}{2}(1–\cos 2\alpha ),{\cos }^2\alpha =\frac{1}{2}(1+\cos 2\alpha ),(\mathrm{降}\mathrm{幂}\mathrm{公}\mathrm{式})\\ & \tan 2\alpha =\frac{2\tan \alpha }{1–{\tan }^2\alpha },\cot 2\alpha =\frac{{\cot }^2\alpha -1}{2\cot \alpha }\end{array}$

• 半角公式：

  $\begin{array}{rl}& {\sin }^2\frac{a}{2}=\frac{1}{2}(1–\cos \alpha ),{\cos }^2\frac{\alpha }{2}=\frac{1}{2}(1+\cos \alpha ),(\mathrm{降}\mathrm{幂}\mathrm{公}\mathrm{式})\\ & \sin \frac{\alpha }{2}=\pm \sqrt{\frac{1–\cos \alpha }{2}},\cos \frac{\alpha }{2}=\pm \sqrt{\frac{1+\cos \alpha }{2}},\\ & \tan \frac{\alpha }{2}=\frac{1–\cos \alpha }{\sin \alpha }=\frac{\sin \alpha }{1+\cos \alpha }=\pm \sqrt{\frac{1–\cos \alpha }{1+\cos \alpha }},\\ & \cot \frac{\alpha }{2}=\frac{\sin \alpha }{1–\cos \alpha }=\frac{1+\cos \alpha }{\sin \alpha }=\pm \sqrt{\frac{1+\cos \alpha }{1-\cos \alpha }}\end{array}.$

• 和差公式：

  $\begin{array}{rl}& \sin (\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta ,\cos (\alpha \pm \beta )=\cos \alpha cos\beta \mp \sin \alpha \cos \beta ,\\ & \tan (\alpha \pm \beta )=\frac{\tan \alpha \pm \tan \beta }{1\mp \tan \alpha \tan \beta },\cot (\alpha \pm \beta )=\frac{\cot \alpha \cot \beta \mp 1}{\cot \beta \pm \cot \alpha }\end{array}$

• 积化和差与和差化积公式：

  ①积化和差公式：

      $\begin{array}{rl}& \sin \alpha \cos \beta =\frac{1}{2}[\sin (\alpha +\beta )+\sin (\alpha –\beta )],\cos \alpha \sin \beta =\frac{1}{2}[\sin (\alpha +\beta )–\sin (\alpha –\beta )],\\ & \cos \alpha \cos \beta =\frac{1}{2}[\cos (\alpha +\beta )+\cos (\alpha –\beta )],\sin \alpha \sin \beta =\frac{1}{2}[\cos (\alpha –\beta )–\cos (\alpha +beta)]\end{array}$

  ②和差化积公式：

  $\begin{array}{rl}& \sin \alpha \sin \beta =2\sin \frac{\alpha +\beta }{2}\cos \frac{\alpha –\beta }{2},\sin \alpha –\sin \beta =2\sin \frac{\alpha –\beta }{2}\cos \frac{\alpha +\beta }{2},\\ & \cos \alpha +cos\beta =2\cos \frac{\alpha +\beta }{2}\cos \frac{\alpha –\beta }{2},\cos \alpha –\cos \beta =-2\sin \frac{\alpha +\beta }{2}\sin \frac{\alpha –\beta }{2}\end{array}$

• 万能公式：

  $\mathrm{若}u=\tan \frac{x}{2}(-\pi <x<\pi ),\mathrm{则}\sin x=\frac{2u}{1+u^2},\cos x=\frac{1–u^2}{1+u^2}.$