Pythagorean Identities

An identity is an equation that is true for all possible values that are substituted in the equation.  A Pythagorean identity, also known as the Pythagorean trigonometric identity, is an identity expressing the Pythagorean Theorem in terms of trigonometric functions.

The 3 Pythagorean trigonometric identities are shown below:

The First Identity

It is the most basic or fundamental Pythagorean identity and is given by the expression:

sin² θ + cos² θ = 1

Pythagorean identities are useful in simplifying trigonometric expressions having trigonometric functions such as sin, cos, and tan.

Let us learn how to derive the fundamental Pythagorean identity.

Proof

Consider a right triangle ABC with side lengths a, b, and c that follows the Pythagorean Theorem

a² + b² = c², here a & b are the two legs, c = hypotenuse

Dividing both sides of the expression by c², we get,

=> a²/c² + b²/c² = /c²/c²

=> (a/c)² + (b/c)² = 1

=> cos² θ + sin² θ = 1 (∵ a/c = base/hypotenuse = cos θ, b/c = perpendicular/hypotenuse = sin θ)

 => sin² θ + cos² θ = 1

The above identity is thus another form of the Pythagorean Theorem and is true for all values of θ

Alternative Method – Using Unit Circle

The above Pythagorean identity can also be obtained using a unit circle, a circle of radius 1 unit (c = 1) we learned that a point on the unit circle is represented by the coordinates (a, b) = (cos θ, sinθ). Thus, the height or perpendicular of the right triangle is sinθ and the base is cos θ as shown below:

If we place the right triangle ABC in a unit circle and apply the Pythagorean Theorem, we get

 a² + b² = c²

=> sin² θ + cos² θ = 1 (∵ a = cos θ, b = sin θ)

The Second Identity

Using the above Pythagorean identity, we can obtain 2 more Pythagorean identities. Let us see how we can obtain them.

As we know, fundamental Pythagorean identity is given by:
cos² θ + sin² θ = 1

Dividing both sides of the equation by cos² θ, we get

=> cos² θ/ cos² θ + sin² θ/ cos² θ = 1/ cos² θ

=> 1 + tan² θ = sec² θ (∵ sin² θ/ cos² θ = tan² θ, 1/ cos² θ = sec² θ)

The above equation is the secondPythagorean identity. Here, it should be noted that there are values of θ for which tangent and secant is undefined:

When cos θ = 0, tan θ = sin θ/cos and sec θ = 1/cos θ are undefined

The Third Identity

Similarly, the third Pythagorean identity is obtained as shown below:
Here, we will again use the fundamental Pythagorean identity

cos² θ + sin² θ = 1

Dividing both sides of the equation by sin² θ, we get

=> cos² θ/ sin² θ + sin² θ/ sin² θ = 1/ sin² θ

=> cot² θ + 1 = cosec² θ

=> 1 + cot² θ = cosec² θ (∵ cos² θ/sin² θ = tan² θ, 1/sin² θ = cosec² θ)

When sin θ = 0, cot θ = cos θ/sin and cosec θ = 1/sin θ are undefined

List

A list of allthePythagorean identities and their variations are given in the table below:

Pythagorean Identities	Variations
sin2 θ + cos2 θ = 1	sin2 θ = 1 – cos2 θ, cos2 θ = 1 – sin2 θ
1 + tan2 θ = sec2 θ	tan2 θ = sec2 θ – 1
1 + cot2 θ = cosec2 θ	cot2 θ = cosec2 θ – 1

Let us learn how to use Pythagorean identities involving solved examples.

Solved Examples