The Pythagorean Theorem gives the relationship between the sides of a right triangle. The following identities show the relationships between the trigonometric functions of a particular angle.
sin2 θ + cos2 θ = 1
tan2 θ + 1 = sec2 θ
1 + cot2 θ = csc2 θ
Why then are these referred to as the Pythagorean Identities?
The reason these identities are called Pythagorean identities is because they are derived from the Pythagorean Theorem. The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
If we consider a unit circle (a circle with a radius of one), the points on the circle can be associated with right triangles by drawing a line from the origin to a point on the circle. The three sides of the right triangle are then the x-coordinate (adjacent side), the y-coordinate (opposite side), and the distance from the origin to the point on the circle (hypotenuse).
Using trigonometric functions, we can express the ratios of the sides of the right triangle. For example, sin θ is equal to the ratio of the opposite side to the hypotenuse, and cos θ is equal to the ratio of the adjacent side to the hypotenuse.
By squaring sin θ and cos θ, and adding them together, we get sin^2 θ + cos^2 θ. Using the Pythagorean Theorem, we know that this sum should be equal to 1 squared (which is just 1) because the hypotenuse of the right triangle is always 1 in a unit circle. Therefore, we have sin^2 θ + cos^2 θ = 1. This is the first Pythagorean identity.
Similarly, by squaring tan θ (the ratio of the opposite side to the adjacent side) and adding 1, we get tan^2 θ + 1. Using the Pythagorean Theorem, we know that this sum should be equal to the hypotenuse squared (which is just 1 squared), giving us tan^2 θ + 1 = 1. Rearranging this equation, we get tan^2 θ + 1 = sec^2 θ. This is the second Pythagorean identity.
Lastly, by squaring cot θ (the ratio of the adjacent side to the opposite side) and adding 1, we get cot^2 θ + 1. Using the Pythagorean Theorem, we know that this sum should be equal to the hypotenuse squared (which is just 1 squared), giving us cot^2 θ + 1 = 1. Rearranging this equation, we get 1 + cot^2 θ = csc^2 θ. This is the third Pythagorean identity.
So, although these identities are not directly related to the lengths of the sides of a right triangle, they are derived from the Pythagorean Theorem and are therefore referred to as Pythagorean identities.