A **trigonometric identity** it is an equality between expressions that involve trigonometric functions and that is true for all the values of the variable (or angle) in which they are defined. From the Pythagorean theorem we can derive the fundamental or basic identities and from these others, generally called auxiliaries.

**Fundamental trigonometric identities **

**Sine-cosine ratio **

cos² α + sin² α = 1

**Secant - tangent ratio**

sec² α = 1 + tg² α

**Crop-cotangent relationship**

csc² α = 1 + ctg² α

**Reciprocal trigonometric identities**

**Cosecant**

csc α = 1 / sin α

**Secant**

sec α = 1 / cos α

**Cotangent**

ctg α = 1 / tg α

**Trigonometric identities of the double angle**

sin 2α = 2 sin · α cos α

Cos 2α = cos² α - sin² α

tg 2α = 2tg α / (1 - tg² α)

**Trigonometric identities of the half angle**

sin (α / 2) = ± √ [(1 - cos α) / 2]

cos (α / 2) = ± √ [(1 + cos α) / 2]

tg (α / 2) = ± √ [(1 - cos α) / (1 + cos α)]

**Operations transformation identities**

** Addition to product step**

sin α + sin β = 2 sin [(α + β) / 2] cos [(α - β) / 2]

sin α - sin β = 2 cos [(α + β) / 2] sin [(α - β) / 2]

cos α + cos β = 2 cos [(α + β) / 2] cos [(α - β) / 2]

cos α - cos β = - 2 sin [(α + β) / 2] sin [(α - β) / 2]

**Step from product to sum**

sin α · cos β = 1/2 [sin (α + β) + cos (α - β)]

cos α · sin β = 1/2 [cos (α + β) + cos (α - β)]

cos α · cos β = 1/2 [cos (α + β) + cos (α - β)]

sin α · sin β = - 1/2 [sin (α + β) - cos (α - β)]

**Demonstration**

In order to demonstrate the trigonometric identities, we will start from a right triangle as shown in the following figure:

If we take the angle α as a reference, the side "a" (AC) will be the opposite leg, "b" (AC) the adjacent or contiguous leg and h (AB, which is the largest side of the three) the hypotenuse.

Considering the above and applying the Pythagorean theorem, we can say the following:

*In any right triangle, the sum of the length of its side “a” (opposite leg) squared, plus the length of the side “b” (adjacent leg) squared, must result in the side h (hypotenuse) squared. *

That is to say:

h² = a² + b² (1)

**First fundamental identity**

First, we must bring the Pythagorean theorem (1) to the value of 1, how? Dividing both sides of the equality by h²:

h² / h² = (a² + b²) / h²

1 = a² / h² + b² / h² (2)

If we now consider the following ratios related by the sides of a right triangle:

sin α = a / h = opposite leg / hypotenuse (3)

cos α = b / h = adjacent leg / hypotenuse (4)

And we substitute them in equation (2) we will have our first trigonometric identity:

**1 = ****sin² α + cos² α** (5)

**Second fundamental identity**

By dividing the first identity (5) by cos² α, we have:

(1 / cos² α) = (sin² α + cos² α) / cos² α

(1 / cos² α) = (sin² α / cos² α) + (cos² α / cos² α)

Remembering that tag α = (sin α / cos α) and sec α = (1 / cos α), we obtain:

**sec****² α**** = tag****² α + 1 ** (6)

**Third fundamental identity**

Considering that csc α = (1 / sinα) and ctg α = (cos α / sin α) and dividing the first identity (5) by sin² α, we have:

(1 / sin² α) = (sin² α + cos² α) / sin² α

(1 / sin² α) = (sin² α / sin² α) + (cos² α / sin² α)

**csc****² α**** = 1 + ctg****² α ** (7)

In general, from the first fundamental identity (5) and by carrying out simple operations, we can find some 24 more identities! Let's remember that all this started from a right triangle!