trig identities

aime

\sin\left(a+b\right)

\sin\left(a\right)\cos\left(b\right)+\cos\left(a\right)\sin\left(b\right)

\sin\left(a-b\right)

\sin\left(a\right)\cos\left(b\right)-\cos\left(a\right)\sin\left(b\right)

\cos(a+b)

\cos(a)\cos(b)-\sin(a)\sin(b)

\cos\left(a-b\right)

\cos(a)\cos(b)+\sin\left(a\right)\sin(b)

\tan\left(a+b\right)

\frac{\tan\left(a\right)+\tan\left(b\right)}{1-\tan\left(a\right)\tan\left(b\right)}

\tan\left(a-b\right)

\frac{\tan\left(a\right)-\tan\left(b\right)}{1+\tan\left(a\right)\tan\left(b\right)}

A\sin\left(x\right)+B\cos\left(x\right)

\sqrt{A^2+B^2}\left(\frac{A\sin x}{\sqrt{A^2+B^2}}+\frac{B\cos x}{\sqrt{A^2+B^2}}\right)

\sin\left(2a\right)

2\sin\left(a\right)\cos\left(a\right)

\cos\left(2a\right)

\cos^2\left(a\right)-\sin^2\left(a\right)=2\cos^2\left(a\right)-1=1-2\sin^2\left(a\right)

\tan\left(2a\right)

\frac{2\tan\left(a\right)}{1-\tan^2\left(a\right)}

1-\cos\left(a\right)

2\sin^2\left(\frac{a}{2}\right)

1+\cos\left(a\right)

2\cos^2\left(\frac{a}{2}\right)

\sin\left(a\right)

\frac{2\tan\left(\frac{a}{2}\right)}{1+\tan^2\left(\frac{a}{2}\right)}

\cos\left(a\right)

\frac{1-\tan^2\left(\frac{a}{2}\right)}{1+\tan^2\left(\frac{a}{2}\right)}

\tan\left(a\right)

\frac{2\tan\left(\frac{a}{2}\right)}{1-\tan^2\left(\frac{a}{2}\right)}

\sin\left(\frac{a}{2}\right)

\pm\sqrt{\frac{1-\cos\left(a\right)}{2}}

\cos\left(\frac{a}{2}\right)

\pm\sqrt{\frac{1+\cos\left(a\right)}{2}}

\tan\left(\frac{a}{2}\right)

\pm\sqrt{\frac{1-\cos\left(a\right)}{1+\cos\left(a\right)}}

\sin\left(3a\right)

3\sin a-4\sin^3a

\cos\left(3a\right)

4\cos^3a-3\cos a

\tan\left(3a\right)

\frac{3\tan a-\tan^3a}{1-3\tan^2a}

\sin\left(60^{}-a\right)\sin\left(a\right)\sin\left(60+a\right)

\frac14\sin\left(3a\right)

\cos\left(60-a\right)\cos\left(a\right)\cos\left(60+a\right)

\frac14\cos\left(3a\right)

\tan\left(60-a\right)\tan\left(a\right)\tan\left(60+a\right)

\tan\left(3a\right)

\sin\left(15\right)

\frac{\sqrt6-\sqrt2}{4}

\cos\left(15\right)

\frac{\sqrt6+\sqrt2}{4}

\sin\left(18\right)

\frac{\sqrt5-1}{4}

\cos\left(36\right)

\frac{\sqrt5+1}{4}

\tan x+\cot x

\frac{2}{\sin\left(2x\right)}

\tan x-\cot x

-2\cot\left(2x\right)

\sin u\cos v

\frac12\left\lbrack\sin\left(u+v\right)+\sin\left(u-v\right)\right\rbrack

\cos u\sin v

\frac12\left\lbrack\sin\left(u+v\right)-\sin\left(u-v\right)\right\rbrack

\cos u\cos v

\frac12\left\lbrack\cos\left(u+v\right)+\cos\left(u-v\right)\right\rbrack

\sin u\sin v

\frac12\left\lbrack\cos\left(u-v\right)-\cos\left(u+v\right)\right\rbrack

\sin x+\sin y

2\sin\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)

\sin x-\sin y

2\cos\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right)

\cos x+\cos y

2\cos\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)

\cos x-\cos y

-2\sin\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right)

\frac{1}{1+\cot\left(x\right)}+\frac{1}{1+\tan\left(x\right)}

1\left(0<x<90\right)

x+y=k\pi+\frac{\pi}{4}

\left(1+\tan x\right)\left(1+\tan y\right)=2

\sin\left(a+b\right)\sin\left(a-b\right)

\sin^2a-\sin^2b=\cos^2b-\cos^2a

\cos\left(a+b\right)\cos\left(a-b\right)

\cos^2a-\sin^2b=\cos^2b-\sin^2a

\tan\left(x+y+z\right)

\frac{\tan x+\tan y+\tan z-\tan x\tan y\tan z}{1-\tan x\tan y-\tan y\tan z-\tan x\tan z}

x+y+z=k\pi

\tan x+\tan y+\tan z=\tan x\tan y\tan z

\cos\left(a\right)\cos\left(2a\right)\cos\left(4a\right)\ldots\cos\left(2^{n}a\right)

\frac{\sin\left(2^{n+1}a\right)}{2^{n+1}\sin\left(a\right)}

\cos\left(a\right)+\cos\left(a+b\right)+\cos\left(a+2b\right)+\cdots+\cos\left(a+\left(n-1\right)b\right)

\frac{\sin\left(a+b\cdot\frac{2n-1}{2}\right)-\sin\left(a-\frac{b}{2}\right)}{2\sin\left(\frac{b}{2}\right)}=\frac{\cos\left(a+b\cdot\frac{n-1}{2}\right)\cdot\sin\left(\frac{n}{2}\cdot b\right)}{\sin\left(\frac{b}{2}\right)}

\sin\left(a\right)+\sin\left(a+b\right)+\sin\left(a+2b\right)+\cdots+\sin\left(a+\left(n-1\right)\cdot b\right)

\frac{\cos\left(a-\frac{b}{2}\right)-\cos\left(a+\frac{2n-1}{2}\cdot b\right)}{2\sin\left(\frac{b}{2}\right)}=\frac{\sin\left(a+\frac{n-1}{2}\cdot b\right)\cdot\sin\left(\frac{n}{2}\cdot b\right)}{\sin\left(\frac{b}{2}\right)}

x^2+y^2=r^2

x=r\cdot\cos\theta,y=r\cdot\sin\theta

\frac{x^2}{a^2}+\frac{y^2}{b^2}=1

x=a\cdot\cos\theta,y=b\cdot\sin\theta

\frac{x^2}{a^2}-\frac{y^2}{b^2}=1

x=a\cdot\sec\theta,y=b\cdot\tan\theta

\frac{1}{1+x^2}

x=\tan\theta\Rightarrow\frac{1}{1+x^2}=\cos^2\theta

\frac{x+y}{1-xy}

x=\tan a,y=\tan b\Rightarrow\frac{x+y}{1-xy}=\tan\left(a+b\right)

\frac{x-y}{1+xy}

x=\tan a,y=\tan b\Rightarrow\frac{x-y}{1+xy}=\tan\left(a-b\right)

\sqrt{a^2-x^2}

x=a\cdot\cos,\sqrt{a^2-x^2}=a\cdot\sin\theta

\sqrt{x^2-a^2}

x=a\cdot\sec\theta,\sqrt{x^2-a^2}=a\cdot\tan\theta

\sqrt{a^2+x^2}

x=a\cdot\tan\theta,\sqrt{a^2+x^2}=a\cdot\sec\theta

\frac{x}{\sqrt{1+x^2}}

x=\tan\theta\Rightarrow\frac{x}{\sqrt{1+x^2}}=\sin\theta

\frac{1}{\sqrt{1+x^2}}

x=\tan\theta\Rightarrow\frac{1}{\sqrt{1+x^2}}=\cos\theta

\frac{2x}{1+x^2}

x=\tan\frac{\theta}{2}\Rightarrow\frac{2x}{1+x^2}=\sin\theta

\frac{1-x^2}{1+x^2}

x=\tan\frac{\theta}{2}\Rightarrow\frac{1-x^2}{1+x^2}=\cos\theta

\frac{2x}{1-x^2}

x=\tan\frac{\theta}{2}\Rightarrow\frac{2x}{1-x^2}=\tan\theta

x^2+y^2+z^2=r^2

x=r\cos a\cos b,y=r\cos a\sin b,z=r\sin a