Trigonometric identities

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:

Fundamental identities

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:

sec² α  = tag² 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)

csc² α  = 1 + ctg² 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!