Tożsamości trygonometryczne

Szczególny przypadek powyższych wzorów to wzory na funkcje kąta podwojonego. Źródło^[4]:

${\displaystyle \sin 2x=2\sin x\cos x}$

${\displaystyle \cos 2x=\cos ^{2}x-\sin ^{2}x=1-2\sin ^{2}x=2\cos ^{2}x-1}$

${\displaystyle \operatorname {tg} 2x={\frac {2\operatorname {tg} x}{1-\operatorname {tg} ^{2}\ x}}}$

${\displaystyle \operatorname {ctg} 2x={\frac {\operatorname {ctg} x-\operatorname {tg} x}{2}}={\frac {\operatorname {ctg} ^{2}\ x-1}{2\operatorname {ctg} x}}}$

Źródło^[5]:

${\displaystyle \sin 3x=3\sin x-4\sin ^{3}x}$

${\displaystyle \cos 3x=4\cos ^{3}x-3\cos x}$

${\displaystyle \operatorname {tg} 3x={\frac {3\operatorname {tg} x-\operatorname {tg} ^{3}\ x}{1-3\operatorname {tg} ^{2}\ x}}}$

${\displaystyle \operatorname {ctg} 3x={\frac {\operatorname {ctg} ^{3}\ x-3\operatorname {ctg} x}{3\operatorname {ctg} ^{2}\ x-1}}}$

Źródło^[5]:

${\displaystyle \sin 4x=8\cos ^{3}x\sin x-4\cos x\sin x=4\cos ^{3}x\sin x-4\cos x\sin ^{3}x}$

${\displaystyle \cos 4x=8\cos ^{4}x-8\cos ^{2}x+1=8\sin ^{4}x-8\sin ^{2}x+1=\cos ^{4}x-6\cos ^{2}x\sin ^{2}x+\sin ^{4}x}$

${\displaystyle \operatorname {tg} 4x={\frac {4\operatorname {tg} x-4\operatorname {tg} ^{3}\ x}{1-6\operatorname {tg} ^{2}\ x+\operatorname {tg} ^{4}\ x}}}$

${\displaystyle \operatorname {ctg} 4x={\frac {\operatorname {ctg} ^{4}\ x-6\operatorname {ctg} ^{2}\ x+1}{4\operatorname {ctg} ^{3}\ x-4\operatorname {ctg} x}}}$

Można je znaleźć przez rekurencyjne stosowanie wzorów na funkcje sumy kątów^[5].

${\displaystyle {\begin{aligned}\sin nx&=\sum _{i=0}^{\infty }(-1)^{i}\cdot {n \choose 2i+1}\cos ^{n-2i-1}x\sin ^{2i+1}x\\[2pt]&=\sum _{i=0}^{\lfloor {\frac {n}{2}}\rfloor }(-1)^{i}\cdot {n \choose 2i+1}\cos ^{n-2i-1}x\sin ^{2i+1}x\\&=n\cos ^{n-1}x\sin x-{n \choose 3}\cos ^{n-3}x\sin ^{3}x+{n \choose 5}\cos ^{n-5}x\sin ^{5}x-{n \choose 7}\cos ^{n-7}x\sin ^{7}x+\dots \end{aligned}}}$

${\displaystyle {\begin{aligned}\cos nx&=\sum _{i=0}^{\infty }(-1)^{i}\cdot {n \choose 2i}\cos ^{n-2i}x\sin ^{2i}x\\[2pt]&=\sum _{i=0}^{\lfloor {\frac {n}{2}}\rfloor }(-1)^{i}\cdot {n \choose 2i}\cos ^{n-2i}x\sin ^{2i}x\\&=\cos ^{n}x-{n \choose 2}\cos ^{n-2}x\sin ^{2}x+{n \choose 4}\cos ^{n-4}x\sin ^{4}x-{n \choose 6}\cos ^{n-6}x\sin ^{6}x+\dots \end{aligned}}}$

${\displaystyle \operatorname {tg} nx=\sum _{i=0}^{\left\lfloor {\frac {n}{2}}\right\rfloor }{n \choose 2i+1}\operatorname {tg} ^{2i+1}x\cdot (-1)^{i}\cdot \left(\sum _{i=0}^{\left\lfloor {\frac {n}{2}}\right\rfloor }{n \choose 2i}\operatorname {tg} ^{2i}x\cdot (-1)^{i}\right)^{-1}}$

Źródła^[7][1][8]:

${\displaystyle \sin x\pm \sin y=2\sin {\frac {x\pm y}{2}}\cdot \cos {\frac {x\mp y}{2}}}$

${\displaystyle \cos x+\cos y=2\cos {\frac {x+y}{2}}\cdot \cos {\frac {x-y}{2}}}$

${\displaystyle \cos x-\cos y=-2\sin {\frac {x+y}{2}}\cdot \sin {\frac {x-y}{2}}}$

${\displaystyle \operatorname {tg} x\pm \operatorname {tg} y={\frac {\sin(x\pm y)}{\cos x\cos y}}}$

${\displaystyle \operatorname {tg} x+\operatorname {ctg} y={\frac {\cos(x-y)}{\cos x\sin y}}}$

${\displaystyle \operatorname {ctg} x\pm \operatorname {ctg} y={\frac {\sin(y\pm x)}{\sin x\sin y}}}$

${\displaystyle \operatorname {ctg} x-\operatorname {tg} y={\frac {\cos(x+y)}{\sin x\cos y}}}$

${\displaystyle 1-\cos x=2\sin ^{2}{\frac {x}{2}}}$

${\displaystyle 1+\cos x=2\cos ^{2}{\frac {x}{2}}}$

${\displaystyle 1-\sin x=2\sin ^{2}\left({\frac {1}{4}}\pi -{\frac {1}{2}}x\right)=2\cos ^{2}\left({\frac {1}{4}}\pi +{\frac {1}{2}}x\right)=\left(\sin {\frac {x}{2}}-\cos {\frac {x}{2}}\right)^{2}}$

${\displaystyle 1+\sin x=2\cos ^{2}\left({\frac {1}{4}}\pi -{\frac {1}{2}}x\right)=2\sin ^{2}\left({\frac {1}{4}}\pi +{\frac {1}{2}}x\right)=\left(\sin {\frac {x}{2}}+\cos {\frac {x}{2}}\right)^{2}}$

Źródła^[1][9]:

${\displaystyle \cos x\cdot \cos y={\frac {\cos(x-y)+\cos(x+y)}{2}}}$

${\displaystyle \sin x\cdot \sin y={\frac {\cos(x-y)-\cos(x+y)}{2}}}$

${\displaystyle \sin x\cdot \cos y={\frac {\sin(x-y)+\sin(x+y)}{2}}}$

${\displaystyle \sin x\cdot \sin y\cdot \sin z={\frac {\sin(x+y-z)+\sin(y+z-x)+\sin(z+x-y)-\sin(x+y+z)}{4}}}$

${\displaystyle \sin x\cdot \sin y\cdot \cos z={\frac {-\cos(x+y-z)+\cos(y+z-x)+\cos(z+x-y)-\cos(x+y+z)}{4}}}$

${\displaystyle \sin x\cdot \cos y\cdot \cos z={\frac {\sin(x+y-z)-\sin(y+z-x)+\sin(z+x-y)+\sin(x+y+z)}{4}}}$

${\displaystyle \cos x\cdot \cos y\cdot \cos z={\frac {\cos(x+y-z)+\cos(y+z-x)+\cos(z+x-y)+\cos(x+y+z)}{4}}}$

${\displaystyle \sin ^{2}x={\frac {1-\cos 2x}{2}}}$

${\displaystyle \cos ^{2}x={\frac {1+\cos 2x}{2}}}$

${\displaystyle \sin ^{2}x-\sin ^{2}y=\sin(x+y)\cdot \sin(x-y)}$

${\displaystyle \sin ^{2}x\cos ^{2}x={\frac {1-\cos 4x}{8}}={\frac {\sin ^{2}2x}{4}}}$

${\displaystyle \sin ^{3}x={\frac {3\sin x-\sin 3x}{4}}}$

${\displaystyle \cos ^{3}x={\frac {3\cos x+\cos 3x}{4}}}$

${\displaystyle \sin ^{4}x={\frac {\cos 4x-4\cos 2x+3}{8}}}$

${\displaystyle \cos ^{4}x={\frac {\cos 4x+4\cos 2x+3}{8}}}$