三角函数恒等变换公式

$$ \begin{align*} \textbf{I: 基本恒等式}\notag \\ \sin^2 x + \cos^2 x &= 1 \\ \tan x &= \frac{\sin x}{\cos x} \\ 1 + \tan^2 x &= \sec^2 x \\ 1 + \cot^2 x &= \csc^2 x \\ \hline \textbf{II: 诱导公式} \notag \\ (奇变偶不变，符号看象限) \notag \\ \sin(x+2k\pi) &= \sin x \\ \cos(x+2k\pi) &= \cos x \\ \tan(x+k\pi) &= \tan x \\ \hdashline \sin(-x) &= -\sin x \\ \cos(-x) &= \cos x \\ \tan(-x) &= -\tan x \\ \hdashline \sin(\pi - x) &= \sin x \\ \cos(\pi - x) &= -\cos x \\ \hdashline \sin(\pi + x) &= -\sin x \\ \cos(\pi + x) &= -\cos x \\ \hdashline \sin(\frac{\pi}{2}-x)&=\cos x \\ \cos(\frac{\pi}{2}-x)&=\sin x \\ \hdashline \sin(\frac{\pi}{2}+x)&=\cos x \\ \cos(\frac{\pi}{2}+x)&=-\sin x \\ \hline \textbf{III: 和差公式} \\ \sin(a\pm b)&=\sin a\cos b\pm\cos a\sin b \\ \cos(a\pm b)&=\cos a\cos b\mp\sin a\sin b \\ \tan(a\pm b)&=\frac{\tan a\pm\tan b}{1\mp\tan a\tan b} \\ \hline \textbf{IV: 二倍角公式} \\ \sin 2x &= 2\sin x\cos x \\ \cos 2x &= \cos^2 x - \sin^2 x \\ &= 1 - 2\sin^2 x \\ &= 2\cos^2 x - 1 \\ \tan 2x &= \frac{2\tan x}{1-\tan^2 x} \\ \hline \textbf{V: 半角公式} \\ \sin^2\frac{x}{2} &= \frac{1-\cos x}{2} \\ \cos^2\frac{x}{2} &= \frac{1+\cos x}{2} \\ \tan\frac{x}{2} &= \frac{\sin x}{1+\cos x} \\ \tan\frac{x}{2} &= \frac{1-\cos x}{\sin x} \\ \tan\frac{x}{2} &= \frac{\sin x}{1+\cos x} \\ \end{align*} $$