# 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:

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

## Solved Examples

If sin 30° = ½. Find the value of cos 30°.

Solution:

As we know,
sin2 θ + cos2 θ = 1
=> sin2 30°+ cos2 30° = 1
=> (1/2)2 + cos2 (30°) = 1 (∵ sin (30°) = ½)
=> cos2 (30°) = 1 – ¼
=> cos2 (30°) = ¾
=> cos(30°) = √¾ = ±√3/2
Since 30° is an acute angle, the value of cos (30°) = √3/2

If θ is in the second quadrant, and sin θ = 3/5, what is the value of cos θ?

Solution:

Substituting the value of sin θ, and solving for cos θ, we get
sin2 θ + cos2 θ = 1
=> (3/5)2 + cos2 θ = 1
=> cos2 θ = 1 – (3/5)2
=> cos2 θ = 25/25 – 9/25
=> cos2 θ = 16/25
=> cosθ = √16/25
=> cosθ = ±4/5
Since the cosine is in the second quadrant, the value of cosθ is -4/5

Simplify the expression: (1 + cos θ)(1 – cos θ)

Solution:

Given,
(1 + cos θ)(1 – cos θ)
=> 1 – cos2 θ
=> sin2 θ