What are Pythagorean Identities?

What are Pythagorean Identities? In mathematics, identity is an equation that is true for all possible values. An equation that involves trigonometric functions and is true for every single value substituted for the variable is called trigonometric identity. We assume both sides are “defined” for that value. Trigonometric identities are especially useful for simplifying trigonometric expressions. The most important trigonometric identities are those involving the Pythagorean Theorem. In this article, we will learn what Pythagorean Identities are.

Pythagorean Identities List

1. sin² θ + cos² θ = 1

2. tan² θ + 1 = sec² θ

3. 1 + cot² θ = cosec² θ

Derivation

Let us see how we derive these equations. Consider the unit circle. A point on the unit circle can be represented by the coordinates (cos θ, sin θ ).

Here, x = cos θ, y = sin θ.

The sides of the right triangle in the unit circle have the values of sin θ and cos θ.

Applying the Pythagorean Theorem, we can write x²+y² = 1

Substitute x and y, we get

sin² θ + cos² θ = 1.

This equation is called a Pythagorean Identity.

It is true for all values of θ in the unit circle.

Using this first Pythagorean Identity, we can find two additional Pythagorean Identities.

sin² θ + cos² θ = 1

Divide each term by cos² θ

(sin² θ/cos² θ )+ (cos² θ/cos² θ) = 1/cos² θ

tan² θ + 1 = sec² θ

This is the second Pythagorean Identity.

Now divide each term of first equation by sin² θ

(sin² θ/sin² θ) + (cos² θ/sin² θ) = 1/sin² θ

1 + cot² θ = cosec² θ

This is the third Pythagorean Identity.

Example 1: In a right-angled triangle ABC,  angle C = 90⁰. BAC = θ, sin θ = 4/5. Find the value of cos θ

Solution:

Here we use the identity sin²θ+cos²θ = 1

(4/5)²+cos²θ = 1

cos θ = √(1-(⅘)² 

=  √(9/25)

= 3/5

Example 2: If sin θ+cos θ = 1/2, what is sin θ.cos θ.

Solution: 

Given sin θ+cos θ = 1/2 

Squaring both sides,

(sin θ+cos θ)² = 1/4

sin² θ+cos²θ+2sin θ cos θ = 1/4

1+2 sin θ cos θ = 1/4

2 sin θ cos θ = (1/4)-1 = -3/4

sin θ cos θ = -3/8.

Also Read

Trigonometry ratios

Trigonometric equations and its solutions