A trigonometric identity is an equality between expressions that involve trigonometric functions and that is true for all angle values (** multiple angles**) in which they are defined.

**Multiple angle identities**

**Trigonometric identities of the double angle**

**Trigonometric identities of the angle ****triple**

**Trigonometric identities of the half angle**

**Demonstration**

**Double angle**

**sin 2α **

In this case, the simplest is to use the identities of the sum of two angles:

doing β = α, we have:

**cos 2α **

For the cosine of the double angle we repeat the same procedure above:

Considering that:

and making β = α:

**tg 2α**

How:

So:

**Triple angle**

To demonstrate the sine and cosine of the triple angle we can do it in two ways. The first is following a procedure similar to the previous one when considering that:

We develop the sum of angles and then the double angle.

Another way is by using the **De Moivre's formula**, which establishes that for any complex number (as well as for any real) "x" and an integer "n", it is verified that:

Where i is the imaginary number:

**sin 3α and ****cos 3α**

Doing n = 3 and x = α in Moivre's formula, we have:

As i² = -1 and i³ = -i:

Equalizing the real and the imaginary part we obtain:

Considering that:

we have:

**tg 3****α**