PYTHAGOREAN IDENTITIES: DEMONSTRATION, EXAMPLE, EXERCISES - MATHS - 2025

Pythagorean identities are all trigonometric equations that hold for any value of the angle and are based on the Pythagorean theorem. The most famous of the Pythagorean identities is the fundamental trigonometric identity:

Sin ² (α) + Cos ² (α) = 1

Figure 1. Pythagorean trigonometric identities.

Next in importance and I use the Pythagorean identity of the tangent and secant:

Tan ² (α) + 1 = Sec ² (α)

And the Pythagorean trigonometric identity involving the cotangent and the cosecant:

1 + Ctg ² (α) = Csc ² (α)

Demonstration

The trigonometric ratios sine and cosine are represented on a circle of radius one (1) known as a trigonometric circle. Said circle has its center at the origin of coordinates O.

Angles are measured from the positive semi-axis of the Xs, for example angle α in figure 2 (see below). Counterclockwise if the angle is positive, and clockwise if it is a negative angle.

The ray with origin O and angle α is drawn, which intercepts the unit circle at point P. Point P is projected orthogonally on the horizontal axis X giving rise to point C. Similarly P is projected perpendicularly on the vertical axis Y giving place to point S.

We have the right triangle OCP at C.

Sine and cosine

It should be remembered that the trigonometric ratio sine is defined on a right triangle as follows:

The sine of an angle of the triangle is the ratio or quotient between the leg opposite the angle and the hypotenuse of the triangle.

Applied to the triangle OCP of figure 2 it would look like this:

Sen (α) = CP / OP

but CP = OS and OP = 1, so that:

Sen (α) = OS

Which means that the projection OS on the Y axis has a value equal to the sine of the displayed angle. It should be noted that the maximum value of the sine of an angle (+1) occurs when α = 90º and the minimum (-1) when α = -90º or α = 270º.

Figure 2. Trigonometric circle showing the relationship between the Pythagorean theorem and the fundamental trigonometric identity. (Own elaboration)

Similarly, the cosine of an angle is the quotient between the leg adjacent to the angle and the hypotenuse of the triangle.

Applied to the triangle OCP of figure 2 it would look like this:

Cos (α) = OC / OP

but OP = 1, so that:

Cos (α) = OC

This means that the projection OC on the X axis has a value equal to the sine of the angle shown. It should be noted that the maximum value of cosine (+1) occurs when α = 0º or α = 360º, while the minimum value of cosine is (-1) when α = 180º.

The fundamental identity

For the right triangle OCP in C, the Pythagorean theorem is applied, which states that the sum of the square of the legs is equal to the square of the hypotenuse:

CP ² + OC ² = OP ²

But it has already been said that CP = OS = Sen (α), that OC = Cos (α) and that OP = 1, so the previous expression can be rewritten as a function of the sine and cosine of the angle:

Sin ² (α) + Cos ² (α) = 1

The axis of the tangent

Just as the X axis in the trigonometric circle is the cosine axis and the Y axis the sine axis, in the same way there is the tangent axis (see figure 3) which is precisely the tangent line to the unit circle at the point B of coordinates (1, 0).

If you want to know the value of the tangent of an angle, the angle is drawn from the positive semi-axis of the X, the intersection of the angle with the axis of the tangent defines a point Q, the length of the segment OQ is the tangent of the angle.

This is because by definition, the tangent of angle α is the opposite leg QB between the adjacent leg OB. That is, Tan (α) = QB / OB = QB / 1 = QB.

Figure 3. The trigonometric circle showing the axis of the tangent and the Pythagorean identity of the tangent. (Own elaboration)

The Pythagorean identity of the tangent

The Pythagorean identity of the tangent can be proved by considering the right triangle OBQ at B (Figure 3). Applying the Pythagorean theorem to this triangle we have that BQ ² + OB ² = OQ ². But it has already been said that BQ = Tan (α), that OB = 1 and that OQ = Sec (α), so that substituting in the Pythagorean equality for the right triangle OBQ we have:

Tan ² (α) + 1 = Sec ² (α).

Example

Check whether or not the Pythagorean identities are fulfilled in the right triangle of legs AB = 4 and BC = 3.

Solution: The legs are known, the hypotenuse needs to be determined, which is:

AC = √ (AB ^ 2 + BC ^ 2) = √ (4 ^ 2 + 3 ^ 2) = √ (16 + 9) = √ (25) = 5.

The angle ∡BAC will be called α, ∡BAC = α. Now the trigonometric ratios are determined:

Sen α = BC / AC = 3/5

Cos α = AB / AC = 4/5

So α = BC / AB = 3/4

Cotan α = AB / BC = 4/3

Sec α = AC / AB = 5/4

Csc α = AC / BC = 5/3

It begins with the fundamental trigonometric identity:

Sin ² (α) + Cos ² (α) = 1

(3/5) ^ 2 + (4/5) ^ 2 = 9/25 + 16/25 = (9 +16) / 25 = 25/25 = 1

It is concluded that it is fulfilled.

- The next Pythagorean identity is that of the tangent:

Tan ² (α) + 1 = Sec ² (α)

(3/4) ^ 2 + 1 = 9/16 + 16/16 = (9 + 16) / 16 = 25/16 = (5/4) ^ 2

And it is concluded that the identity of the tangent is verified.

- In a similar way that of the cotangent:

1 + Ctg ² (α) = Csc ² (α)

1+ (4/3) ^ 2 = 1 + 16/9 = 25/9 = (5/3) ^ 2

It is concluded that it is also fulfilled, with which the task of verifying the Pythagorean identities for the given triangle has been completed.

Solved exercises

Prove the following identities, based on the definitions of the trigonometric ratios and the Pythagorean identities.

Exercise 1

Prove that Cos ² x = (1 + Sin x) (1 - Sin x).

Solution: In the right side we recognize the remarkable product of the multiplication of a binomial by its conjugate which, as we know, is a difference of squares:

Cos ² x = 1 ² - Sin ² x

Then the term with sine on the right side passes to the left side with the sign changed:

Cos ² x + Sen ² x = 1

Noting that the fundamental trigonometric identity has been reached, so it is concluded that the given expression is an identity, that is, it is true for any value of x.

Exercise 2

Starting from the fundamental trigonometric identity and using the definitions of the trigonometric ratios, demonstrate the Pythagorean identity of the cosecant.

Solution: The fundamental identity is:

Sin ² (x) + Cos ² (x) = 1

Both members are divided by Sen ² (x) and the denominator is distributed in the first member:

Sin ² (x) / Sin ² (x) + Cos ² (x) / Sin ² (x) = 1 / Sin ² (x)

It is simplified:

1 + (Cos (x) / Sen (x)) ^ 2 = (1 / Sen (x)) ^ 2

Cos (x) / Sen (x) = Cotan (x) is a (non-Pythagorean) identity that is verified by the very definition of the trigonometric ratios. The same happens with the following identity: 1 / Sen (x) = Csc (x).

Finally you have to:

1 + Ctg ² (x) = Csc ² (x)

References

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6. Miller, Heeren, & Hornsby. (2006). Mathematics: Reasoning And Applications (Tenth Edition). Pearson Education.
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8. Wikipedia. Trigonometry identities and formulas. Recovered from: es.wikipedia.com