第三章 三角恒等变换

两角和与差的正弦、余弦、正切公式：
\(\cos (\alpha \pm \beta ) = \cos a\cos \beta \mp\sin \alpha \sin \beta\)
\(\sin (\alpha \pm \beta ) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta\)
\(\tan (\alpha \pm \beta )=\frac{\tan \alpha \pm \tan \beta }{1\mp \tan \alpha \tan \beta }\)

二倍角的正弦、余弦、正切公式
正弦形式:
\(\sin 2\alpha = 2\sin \alpha \cos \alpha\)
推导过程：
\(\sin 2\alpha =\sin (\alpha +\alpha )=\sin \alpha \cos \alpha +\cos \alpha \sin \alpha =2\sin \alpha \cos \alpha\)

余弦形式:
\(\cos 2\alpha =2\cos^{2}\alpha -1=1-2\sin^{2} \alpha =\cos^{2} \alpha -\sin^{2}\alpha =\frac{1-\tan^{2}\alpha}{1+\tan^{2}\alpha}\)
推导过程：
\(\cos 2\alpha =\cos (\alpha +\alpha )=\cos \alpha \cos \alpha -\sin \alpha \sin \alpha =\cos^{2}\alpha -\sin^{2} \alpha =2\cos^{2} \alpha -1 =1-2\sin^{2}a\)

正切形式：
\(\tan 2\alpha =\frac{2\tan \alpha }{1-\tan^{2}\alpha } =\frac{2\cot \alpha }{\cot^{2}\alpha -1 }=\frac{2}{\cot \alpha -\tan \alpha }\)
推导过程：
\(\tan 2\alpha=\frac{\sin 2\alpha }{\cos 2\alpha } = \frac{2\sin \alpha \cos \alpha}{\cos^{2}\alpha-\sin^{2}\alpha}=\frac{\frac{2\sin a\cos a}{\cos^{2}\alpha } }{\frac{\cos^{2}\alpha-\sin^{2} \alpha }{\cos^{2} \alpha } } =\frac{2\tan \alpha }{1-\tan^{2}\alpha }\)