Fundamental General Identities Involving Trigonometric Ratios

  1. Si{n^2}\theta + Co{s^2}\theta = 1

  2. 1 + Ta{n^2}\theta = Se{c^2}\theta

  3. 1 + Co{t^2}\theta = Cose{c^2}\theta

  4. Sin(\alpha + \beta ) = Sin\alpha Cos\beta  + Cos\alpha Sin\beta

  5. Sin(\alpha - \beta ) = Sin\alpha Cos\beta - Cos\alpha Sin\beta

  6. Cos(\alpha + \beta ) = Cos\alpha Cos\beta - Sin\alpha Sin\beta

  7. Cos(\alpha - \beta ) = Cos\alpha Cos\beta + Sin\alpha Sin\beta

  8. Tan(\alpha + \beta ) = \frac{{Tan\alpha + Tan\beta }}{{1 - Tan\alpha Tan\beta }}

  9. Tan(\alpha - \beta ) = \frac{{Tan\alpha - Tan\beta }}{{1 + Tan\alpha Tan\beta }}

  10. Tan({45^ \circ } + \theta ) = \frac{{1 + Tan\theta }}{{1 - Tan\theta }}

  11. Tan({45^ \circ } - \theta ) = \frac{{1 - Tan\theta }}{{1 + Tan\theta }}

  12. Cot(\alpha + \beta ) = \frac{{Cot\alpha Cot\beta - 1}}{{Cot\alpha + Cot\beta }}

  13. Cot(\alpha - \beta ) = \frac{{Cot\alpha Cot\beta + 1}}{{Cot\alpha - Cot\beta }}

  14. Sin2\theta     = 2Sin\theta Cos\theta

  15. Cos2\theta = Co{s^2}\theta -  Si{n^2}\theta

  16. Cos2\theta = 1 - 2Si{n^2}\theta

  17. Cos2\theta = 2Co{s^2}\theta - 1

  18. Tan2\theta = \frac{{2Tan\theta }}{{1 - Ta{n^2}\theta }} = \frac{{2Cot\theta }}{{Co{t^2}\theta - 1}}

  19. Cot2\theta = \frac{{Co{t^2}\theta -  1}}{{2Cot\theta }} = \frac{{Cot\theta - Tan\theta }}{2} = \frac{{1 - Ta{n^2}\theta }}{{2Tan\theta }}

  20. Sin\frac{\theta }{2} = \pm \sqrt {\frac{{1 - Cos\theta }}{2}}

  21. Cos\frac{\theta }{2} = \pm \sqrt {\frac{{1 + Cos\theta }}{2}}

  22. 1 - Cosn\theta = 2Si{n^2}\frac{{n\theta }}{2}

  23. 1 + Cosn\theta = 2Co{s^2}\frac{{n\theta }}{2}

  24. Tan\frac{\theta }{2} = \frac{{1 - Cos\theta }}{{Sin\theta }} = \frac{{Sin\theta }}{{1 + Cos\theta }} = \pm \sqrt {\frac{{1 - Cos\theta }}{{1 +  Cos\theta }}}

  25. Sin3\theta = 3Sin\theta - 4Si{n^3}\theta

  26. Cos3\theta = 4Co{s^3}\theta - 3Cos\theta

  27. Tan3\theta = \frac{{3Tan\theta - Ta{n^3}\theta }}{{1 - 3Ta{n^2}\theta }}

  28. Cot3\theta = \frac{{Co{t^3}\theta - 3Cot\theta }}{{3Co{t^2}\theta - 1}}