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. \[Cot3\theta = \frac{{Co{t^3}\theta – 3Cot\theta }}{{3Co{t^2}\theta – 1}}\]