goniometric formulas

The word goniometric (from Greek $\gamma\omega\nu$ í$\alpha$ ``angle'' and $\mu\varepsilon\tau\varrho\iota\kappa$ ó$\varsigma$ ``measuring'') concerns the trigonometric functions and their mutual connections. There are a great amount of formulas involving these functions (usually for real arguments).

1. Pythagorean identities
   • $\sin^2{x}+\cos^2{x} = 1$
   • $\tan^2{x}+1 = \sec^2{x}$
   • $1+\cot^2{x} = \csc^2{x}$
2. Fractional identities
   • $\displaystyle \tan{x} = \frac{\sin{x}}{\cos{x}}$
   • $\displaystyle \cot{x} = \frac{\cos{x}}{\sin{x}}$
   • $\displaystyle \cot{x} = \frac{1}{\tan{x}}$
   • $\displaystyle \tan{x} = \frac{1}{\cot{x}}$
   • $\displaystyle \csc{x} = \frac{1}{\sin{x}}$
   • $\displaystyle \sec{x} = \frac{1}{\cos{x}}$
3. Formulas involving radicals
   • $\displaystyle \sin{x} = \pm\frac{\tan{x}}{\sqrt{1+\tan^2{x}}}$
   • $\displaystyle \cos{x} = \pm\frac{1}{\sqrt{1+\tan^2{x}}}$
4. Weierstrass substitution formulas and related formula for $\tan x$
   • $\displaystyle \sin{x} = \frac{\displaystyle 2\tan\left( \frac{x}{2} \right)}{\displaystyle 1+\tan^2\left( \frac{x}{2} \right)}$
   • $\displaystyle \cos{x} = \frac{\displaystyle 1-\tan^2\left( \frac{x}{2} \right)}{\displaystyle 1+\tan^2\left( \frac{x}{2} \right)}$
   • $\displaystyle \tan{x} = \frac{\displaystyle 2\tan\left( \frac{x}{2} \right)}{\displaystyle 1-\tan^2\left( \frac{x}{2} \right)}$
5. Trigonometric functions of a purely imaginary number
   • $\sin(ix)=i\sinh x$
   • $\cos(ix)=\cosh x$
   • $\tan(ix)=i\tanh x$
   • $\cot(ix)=i\coth x$
   • $\csc(ix)=i\operatorname{csch}x$
   • $\sec(ix)=\operatorname{sech}x$
6. Addition formulas and subtraction formulas
   • $\sin(x \pm y) = \sin{x}\cos{y}\pm\cos{x}\sin{y}$
   • $\cos(x \pm y) = \cos{x}\cos{y}\mp\sin{x}\sin{y}$
   • $\displaystyle \tan(x \pm y) = \frac{\tan{x}\pm\tan{y}}{1\mp\tan{x}\tan{y}}$
7. Formulas for trigonometric functions of a complex number
   • $\sin(x+iy) = \sin x\cosh y+i\cos x\sinh y$
   • $\cos(x+iy) = \cos x\cosh y-i\sin x\sinh y$
   • $\displaystyle \tan(x+iy) = \frac{\tan x+i\tanh y}{1-i\tan x\tanh y}$
8. Complement formulas
   • $\displaystyle \sin\left(\frac{\pi}{2}-x\right) = \cos{x}$
   • $\displaystyle \cos\left(\frac{\pi}{2}-x\right) = \sin{x}$
   • $\displaystyle \tan\left(\frac{\pi}{2}-x\right) = \cot{x}$
9. Supplement formulas
   • $\sin(\pi-x) = \sin{x}$
   • $\cos(\pi-x) = -\cos{x}$
   • $\tan(\pi-x) = -\tan{x}$
10. Explement formulas
    • $\sin(2\pi-x) = -\sin{x}$
    • $\cos(2\pi-x) = \cos{x}$
    • $\tan(2\pi-x) = -\tan{x}$
11. Opposite angle formulas
    • $\sin(-x) = -\sin{x}$
    • $\cos(-x) = \cos{x}$
    • $\tan(-x) = -\tan{x}$
12. Periodicity formulas
    • $\sin(x+2\pi) = \sin{x}$
    • $\cos(x+2\pi) = \cos{x}$
    • $\tan(x+\pi) = \tan{x}$
13. Double angle formulas
    • $\sin(2x) = 2\sin{x}\cos{x}$
    • $\cos(2x) = \cos^2{x}-\sin^2{x} = 2\cos^2{x}-1 = 1-2\sin^2{x}$
    • $\displaystyle \tan(2x) = \frac{2\tan{x}}{1-\tan^2{x}}$
14. Triple angle formulas
    • $\sin(3x) = 3\sin{x}-4\sin^3{x} = (4\cos^2{x}-1)\sin{x}$
    • $\cos(3x) = 4\cos^3{x}-3\cos{x} = (1-4\sin^2{x})\cos{x}$
    • $\displaystyle \tan(3x) = \frac{3\tan{x}-\tan^3{x}}{1-3\tan^2{x}}$
15. Half angle formulas
    • $\displaystyle \sin\left(\frac{x}{2}\right) = \pm\sqrt{\frac{1-\cos{x}}{2}}$
    • $\displaystyle \cos\left(\frac{x}{2}\right) = \pm\sqrt{\frac{1+\cos{x}}{2}}$
    • $\displaystyle \tan\left(\frac{x}{2}\right) = \frac{\sin{x}}{1+\cos{x}} = \frac{1-\cos{x}}{\sin{x}} = \pm\sqrt{\frac{1-\cos{x}}{1+\cos{x}}}$
16. Prosthaphaeresis formulas
    • $\displaystyle \sin{x}+\sin{y} = 2\sin\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)$
    • $\displaystyle \sin{x}-\sin{y} = 2\sin\left(\frac{x-y}{2}\right)\cos\left(\frac{x+y}{2}\right)$
    • $\displaystyle \cos{x}+\cos{y} = 2\cos\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)$
    • $\displaystyle \cos{x}-\cos{y} = -2\sin\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right)$
17. Product formulas
    • $\displaystyle \sin{x}\,\sin{y} = \frac{\cos(x-y)-\cos(x+y)}{2}$
    • $\displaystyle \cos{x}\,\sin{y} = \frac{\sin(x+y)-\sin(x-y)}{2}$
    • $\displaystyle \cos{x}\,\cos{y} = \frac{\cos(x-y)+\cos(x+y)}{2}$
18. Other sums and differences
    • $\displaystyle \tan{x}\pm\tan{y} = \frac{\sin(x \pm y)}{\cos{x}\,\cos{y}}$
    • $\displaystyle \cot{x}\pm\cot{y} = \frac{\sin(y \pm x)}{\sin{x}\,\sin{y}}$
    • $\displaystyle \cos{x}\pm\sin{x} = \sqrt{2}\sin\!\left(\frac{\pi}{4}\pm x\right) = \sqrt{2}\cos\!\left(\frac{\pi}{4}\mp x\right)$
19. Linearization formulas
    • Second power
      • $\displaystyle \sin^2{x} = \frac{1-\cos(2x)}{2}$
      • $\displaystyle \cos^2{x} = \frac{1+\cos(2x)}{2}$
      • $\displaystyle \tan^2{x} = \frac{1-\cos(2x)}{1+\cos(2x)}$
    • Third power
      • $\displaystyle \sin^3{x} = \frac{3\sin{x}-\sin(3x)}{4}$
      • $\displaystyle \cos^3{x} = \frac{3\cos{x}+\cos(3x)}{4}$
      • $\displaystyle \tan^3{x} = \frac{3\sin{x}-\sin(3x)}{3\cos{x}+\cos(3x)}$
    • Fourth power
      • $\displaystyle \sin^4{x} = \frac{\cos(4x)-4\cos(2x)+3}{8}$
      • $\displaystyle \cos^4{x} = \frac{\cos(4x)+4\cos(2x)+3}{8}$
      • $\displaystyle \tan^4{x} = \frac{\cos(4x)-4\cos(2x)+3}{\cos(4x)+4\cos(2x)+3}$
20. Recursion formulas
    • $\displaystyle\sin[(n\!+\!1)x] = 2\cos{x}\,\sin(nx)-\sin[(n\!-\!1)x]$
    • $\displaystyle\cos[(n\!+\!1)x] = 2\cos{x}\,\cos(nx)-\cos[(n\!-\!1)x]$
21. Exponential formulas
    • $e^{ix}=\cos x+i\sin x$
    • $e^{-ix}=\cos x-i\sin x$
    • $\displaystyle \cos x=\frac{e^{ix}+e^{-ix}}{2}$
    • $\displaystyle \sin x=\frac{e^{ix}-e^{-ix}}{2i}$
    • $\displaystyle \tan x=\frac{e^{ix}-e^{-ix}}{i(e^{ix}+e^{-ix})}$
22. Some special formulas
    • $\displaystyle\tan{\left(x\!+\!\frac{\pi}{4}\right)} = \frac{\cos{x}+\sin{x}}{\cos{x}-\sin{x}} = \pm\sqrt{\frac{1+\sin{2x}}{1-\sin{2x}}}$
    • $\displaystyle\tan x+\sec x = \tan\left(\frac{x}{2}+\frac{\pi}{4}\right)$
    • $\displaystyle\tan\left(\frac{x\pm y}{2}\right) = \frac{\sin{x}\pm\sin{y}}{\cos{x}+\cos{y}} = \frac{\cos{y}-\cos{x}}{\sin{x}\mp\sin{y}}$
    • $\displaystyle\tan\left(\frac{x+y}{2}\right)\,\tan\left(\frac{x-y}{2}\right) = \frac{\cos{y}-\cos{x}}{\cos{y}+\cos{x}}$
    • $\displaystyle\sin(x+y)\,\sin(x-y) = \sin^2x-\sin^2y$