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：

  \( \begin{align}& \sin 2\alpha = 2\sin \alpha \cos \alpha , \cos 2 \alpha =\cos ^2 \alpha – \sin ^2  \alpha = 1- 2\sin ^2 \alpha = 2 \cos ^2 \alpha -1 \\ & \sin 3 \alpha = -4 \sin ^3 \alpha + 3 \sin \alpha, \cos 3\alpha = 4\cos ^3 \alpha – 3\cos \alpha \\ & \sin ^2 \alpha = \frac{1}{2}( 1 – \cos 2 \alpha), \cos ^2 \alpha = \frac{1}{2}(1 + \cos  2\alpha),(降幂公式) \\ & \tan 2\alpha = \frac{2 \tan \alpha}{ 1 – \tan ^2 \alpha} , \cot 2\alpha = \frac{\cot ^2 \alpha -1}{2 \cot \alpha}\end{align}\)

• 半角公式：

  \(\begin{align} & \sin ^2 \frac{a}{2} = \frac{1}{2}(1 – \cos \alpha), \cos ^ 2 \frac{\alpha}{2} = \frac{1}{2}(1 + \cos \alpha),(降幂公式)\\ & \sin \frac{\alpha}{2} = \pm \sqrt{\frac{1 – \cos \alpha}{2}}, \cos \frac{\alpha}{2} = \pm \sqrt{\frac{1 + \cos \alpha}{2}},\\ & \tan \frac{\alpha}{2} = \frac{1 – \cos \alpha}{\sin \alpha} = \frac{\sin \alpha}{1 + \cos \alpha} = \pm \sqrt{\frac{1 – \cos \alpha}{1 + \cos \alpha}}, \\ & \cot \frac{\alpha}{2} = \frac{\sin \alpha}{1 – \cos \alpha} = \frac{1 + \cos \alpha}{\sin \alpha} = \pm \sqrt{\frac{1+\cos \alpha}{1- \cos \alpha}}\end{align}.\)

• 和差公式：

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

• 积化和差与和差化积公式：

  ①积化和差公式：

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

  ②和差化积公式：

  \(\begin{align} & \sin \alpha \sin \beta = 2 \sin{\frac{\alpha + \beta}{2}} \cos{\frac{\alpha – \beta}{2}}, \sin \alpha – \sin \beta = 2\sin{\frac{\alpha – \beta}{2}} \cos{\frac{\alpha + \beta}{2}},\\ & \cos \alpha + cos \beta = 2 \cos{\frac{\alpha + \beta}{2}}\cos{\frac{\alpha – \beta}{2}},\cos \alpha – \cos \beta = -2\sin{\frac{\alpha + \beta}{2}}\sin{\frac{\alpha – \beta}{2}} \end{align}\)

• 万能公式：

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