Pythagorean Identities

Pythagorean identities, as the name suggests, are derived from the Pythagoras theorem. According to this theorem, in any right-angled triangle, the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides (legs). This theorem can be applied to trigonometric ratios (as they are defined for a right-angled triangle) that results in Pythagorean identities.

Let us learn more about Pythagorean identities along with their proof, examples, and more practice problems.

Pythagorean identities are important identities in trigonometry that are derived from the Pythagoras theorem. These identities are used in solving many trigonometric problems where one trigonometric ratio is given and the other ratios are to be found. The fundamental Pythagorean identity gives the relation between sin and cos and it is the most commonly used Pythagorean identity which says:

• sin²θ + cos²θ = 1 (which gives the relation between sin and cos)

There are other two Pythagorean identities that are as follows:

• sec²θ - tan²θ = 1 (which gives the relation between sec and tan)
• csc²θ - cot²θ = 1 (which gives the relation between csc and cot)

Pythagorean Trig Identities

All Pythagorean trig identities are mentioned below together.

Each of them can be written in different forms by algebraic operations. i.e., each Pythagorean identity can be written in 3 forms as follows:

• sin²θ + cos²θ = 1 ⇒ 1 - sin²θ = cos² θ ⇒ 1 - cos²θ = sin²θ
• sec²θ - tan²θ = 1 ⇒ sec²θ = 1 + tan²θ ⇒ sec²θ - 1 = tan²θ
• csc²θ - cot²θ = 1 ⇒ csc²θ = 1 + cot²θ ⇒ csc²θ - 1 = cot²θ

We are going to prove the Pythagorean identities using the Pythagoras theorem. Let us consider a right-triangle ABC that is right-angled at C. Then AB is the hypotenuse. Let us assume that AB = c, BC = a, and CA = b for our convenience. Let us assume that the angle at B is θ.

In the above figure:

• The opposite side (of θ) = b
• The adjacent side (of θ) = a
• The hypotenuse = c

Let us first define all trigonometric ratios which are further useful in deriving Pythagorean identities in trigonometry.

• sin θ = (opposite)/(hypotenuse) = b / c ⇒ csc θ = c / b
• cos θ = (adjancent)/(hypotenuse) = a / c ⇒ sec θ = c / a
• tan θ = (opposite) / (adjancent) = b / a ⇒ cot θ = a / b

Let us prove each pythagorean trig identity one by one.

Proof of Pythagorean Identity sin²θ + cos²θ = 1

Applying the Pythagoras theorem to the triangle, we get

a² + b² = c²

Dividing each term on both sides by c²,

a² / c² + b² / c² = c² / c²

(a / c)² + (b / c)² = 1

(cos θ)² + (sin θ)² = 1 (or)

sin²θ + cos²θ = 1

Hence proved.

Proof of Pythagorean Identity sec²θ - tan²θ = 1

Again, by Pythagoras theorem

a² + b² = c²

Dividing each term on both sides by a²,

a² / a² + b² / a² = c² / a²

1 + (b / a)² = (c / a)²

1 + (tan θ)² = (sec θ)² (or)

sec²θ - tan²θ = 1

Hence proved.

Proof of Pythagorean Identity csc²θ - cot²θ = 1

By Pythagoras theorem,

a² + b² = c²

Dividing each term on both sides by b²,

a² / b² + b² / b² = c² / b²

(a / b)² + 1 = (c / b)²

(cot θ)² + 1 = (csc θ)² (or)

csc²θ - cot²θ = 1

Hence proved.

• Pythagorean identities are used to prove other trigonometric identies.
  Example: Prove the identity sin⁴x - cos⁴x = sin²x - cos²x.
  Solution:
  We can write
  LHS = sin⁴x - cos⁴x = (sin²x)² - (cos²x)²
  Using a² - b² formula,
  = (sin²x - cos²x) (sin²x + cos²x)
  Using Pythagorean identities, sin²x + cos²x = 1.
  = (sin²x - cos²x) (1)
  = sin²x - cos²x
  = RHS
  Hence proved.
• Pythagorean identities are useful in solving the problems related to heights and distances.
• Pythagorean identities are used to find any trigonometric ratio when another trigonometric ratio is given.
  Example: Find cos x when sin x = 3/5 and x is in the 1^st quadrant.
  Solution:
  From Pythagorean identities,
  cos²x = 1 - sin²x
  cos x = ±√1 - sin²x
  = ±√1 - (3/5)²
  = ±√1 - (9/25)
  = ±√16/25
  = ± 4/5
  Since x is in the first quadrant, cos x is positive. So cos x = 4 / 5.

Related Topics:

• Half-Angle Formulas
• Tangent Formulas
• Double-Angle Formulas
• Trigonometry Formulas
• Cosine Formulas
• Trigonometric Functions

List all Pythagorean Identities.

Here are the 3 Pythagorean identities. Each identity can be written in alternative ways as shown.

How to Prove Pythagorean Identities in Trigonometry?

To prove Pythagorean identities, we use the Pythagoras theorem. For example, if ABC is a right-triangle which is right-angled at B and x is the angle at A, then:

AB² + BC² = AC² ... (1)
Dividing both sides by AC²,
(AB/AC)² + (BC/AC)² = 1
sin²x + cos²x = 1

Similarly, by dividing both sides of (1) by AB² and BC², we can derive the other two Pythagorean identities.

What is the Pythagorean Identities List?

Here is the Pythagorean identities list:

• sin²x + cos²x = 1
• 1 + tan²x = sec²x
• 1 + cot²x = csc²x

How many Pythagorean Identities do We Have?

We have 3 Pythagorean identities in trigonometry. They are as follows:

• sin²θ + cos²θ = 1
• sec²θ - tan²θ = 1
• csc²θ - cot²θ = 1

How do You Use Pythagorean Identities?

The Pythagorean identities are used to prove other trigonometric identities, find the value of a trigonometric ratio by using any other trigonometric ratio, and to solve the problems related to heights and distances.

Are Pythagorean Identities Derived From Pythagorean Theorem?

Yes, Pythagorean identities are derived from Pythagorean theorem. To see the proof of Pythagorean identities, click here.