Trigonometric identities 2

Double angle
\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
\tan(2\alpha)=\frac{2\tan\alpha}{1-\tan^2\alpha}

Triple angle
\sin(3\alpha)=3\sin\alpha-4\sin^3\alpha
\cos(3\alpha)=4\cos^3\alpha-3\cos\alpha
\tan(3\alpha)=\frac{3\tan\alpha-\tan^3\alpha}{1-3\tan^2\alpha}

Half angle
\sin(\frac{\alpha}{2})=\sqrt{\frac{1-\cos\alpha}{2}}
\cos(\frac{\alpha}{2})=\sqrt{\frac{1+\cos\alpha}{2}}
\tan(\frac{\alpha}{2})=\sqrt{\frac{1-\cos\alpha}{1+\cos\alpha}}

To Tangent half angle
t=\tan(\frac{\alpha}{2})
\sin\alpha=\frac{2t}{1+t^2}
\cos\alpha=\frac{1-t^2}{1+t^2}
\tan\alpha=\frac{2t}{1-t^2}

Product-to-sum identities
2\sin\alpha\cos\beta=\sin(\alpha-\beta)+\sin(\alpha+\beta)
2\cos\alpha\cos\beta=\cos(\alpha-\beta)+\cos(\alpha+\beta)
2\sin\alpha\sin\beta=\cos(\alpha-\beta)-\cos(\alpha+\beta)

Sum-to-product identities
\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})