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We find many sources on the internet that provides a table of trigonometric identities (like http://www.fis.ufba.br/~luciano.abreu/TABELA.pdf), but I'd like to know if there is a way to determine these identities instead of just memorizing the entire table.

Thanks

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    $\begingroup$ Did you really mean to post this on a site about Mathematica software? By the way, there is an infinity of trigonometric identities. They can all be determined from a very small number of them in standard ways, such as a statement of the first order linear ODE satisfied by $\cos(x) + i \sin(x)$ (together with their initial conditions) and the definitions of the other trig functions in terms of them, or the power series definition of $\exp(i x)$, or the definition of $\exp(z)$ as the inverse of $\int_1^z \frac{dz}{z}$, or the infinite product representation of $\csc(z)$, etc. $\endgroup$ – whuber Mar 19 '13 at 4:13
  • $\begingroup$ Sorry, I've posted to the wrong stackexchange site. $\endgroup$ – Juliano Nunes Silva Oliveira Mar 19 '13 at 4:40
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There are some of the identities, which are just easy(and necessary) enough that you MUST memorize them. Some of the examples may be(not restricted to):

  1. $$ \sin^2 \theta + \cos^2 \theta = 1 $$
  2. $$ \tan \theta = \dfrac{\sin \theta}{\cos \theta} $$
  3. $$ \cot \theta = \dfrac{1}{\tan \theta} $$
  4. $$ \sin 2\theta = 2 \sin \theta \cdot \cos \theta $$

and some more. All others can always be derived from other properties.

A point to note here is that all the generalized questions/expressions using trigonometric functions are always some kind of identities as they will always be true, no matter what conditions are applied(in co-ordinate geometry).

For eg. suppose we want to prove the identity $ 1 + \cot^2 \theta = \csc^2 \theta $, we can proceed as follows:

$$ \begin{align} 1 + \cot^2 \theta &= 1 + \left( \dfrac{\cos^2 \theta}{\sin \theta} \right)^2 \\ &= \dfrac{\sin^2 \theta + \cos^2 \theta}{\sin^2 \theta} \\ &= \dfrac{1}{\sin^2 \theta} \\ &= \csc^2 \theta \end{align} $$ For some more complicated(or basic) trigonometric identities, you might need geometric(or Euler's representation of co-ordinates($r \cdot e^{\iota \theta}$ form)).

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Here are a few that you need to have on your finger tips.

You need to remember the basic trigonometric functions $\sin$,$\cos$ and $\tan$.The remaining 3 are inverse of the above. $\sin^2 \theta+\cos^2\theta=1$

Memorize the sum of angles property.

$\sin(A \pm B)=\sin A \cos B \pm \cos A \sin B$

$\cos(A \pm B)=\cos A \cos B \mp \sin A \sin B$

and that is necessary because every other identity of the sort $\pi-\theta$ can be derived by just plugging in A as $\pi$, and also solves all other identities of sort.

Courtesy: Crusades of Answers!

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