Pythagorean Identities

Use reciprocal and quotient relationships in trig identities. Grade 12 Math 30-1.

Lesson 5.2 of Trigonometric Identities in Math 30-1: Alberta curriculum lessons.

What you'll learn in this lesson

• The Pythagorean Identities. Discover where the three Pythagorean identities come from, learn all their equivalent forms, and see how dividing by sin²θ or cos²θ derives the second and third identities.
• Simplifying with Pythagorean Identities. Use reciprocal, quotient, and Pythagorean identities together to simplify expressions, through direct substitution, combining fractions, factoring, conjugates, and numerical verification.

Inside the lesson: a free preview

Where Do They Come From?

Starting Point: The Circle Equation. Recall from the previous unit that for any point P(x, y) on a circle with radius r:

x^2 + y^2 = r^2

When we work with the unit circle where r = 1, and substitute x = θ and y = θ:

The Fundamental Identity. ^2θ + ^2θ = 1

This is the foundation of all three Pythagorean identities. Every other Pythagorean identity is derived from this one by dividing through by ^2θ or ^2θ.

The Three Pythagorean Identities

There are three Pythagorean identities. Each one has equivalent rearrangements that are equally valid and useful. You should be comfortable recognising and using all forms.

Identity 1, On Formula Sheet. ^2θ + ^2θ = 1

Identity 2, On Formula Sheet. 1 + ^2θ = ^2θ

Identity 3, On Formula Sheet. 1 + ^2θ = ^2θ

How Are the Second and Third Identities Derived?

Both identities 2 and 3 are derived by dividing every term of identity 1 by either ^2θ or ^2θ. The quotient and reciprocal identities from lesson 5.1 then transform the resulting fractions into the familiar trig ratios.

Deriving Identity 2: Divide every term by cos²θ. Start with identity 1 and divide every term by ^2θ:

(^2θ)/(^2θ) + (^2θ)/(^2θ) = (1)/(^2θ)

Each fraction simplifies using quotient and reciprocal identities:

^2θ + 1 = ^2θ

Why each fraction simplifies:.

Deriving Identity 3: Divide every term by sin²θ. Start with identity 1 and divide every term by ^2θ:

(^2θ)/(^2θ) + (^2θ)/(^2θ) = (1)/(^2θ)

Each fraction simplifies:

1 + ^2θ = ^2θ

Why each fraction simplifies:.

Key Takeaway. All three Pythagorean identities come from the single equation x^2 + y^2 = r^2 applied to the unit circle. The second and third are just the first identity divided by ^2θ and ^2θ respectively.

Worked example: Direct Substitution

Simplify: ^2θθ + ^2θθ. ^2θθ + ^2θθ

Step 1: Factor out cos θ. Both terms contain θ as a factor, so factor it out:

= θ(^2θ + ^2θ)

Step 2: Apply Pythagorean Identity 1. The bracket is exactly identity 1, replace ^2θ + ^2θ with 1:

= θ(^2θ + ^2θ_= 1) = θ(1) = θ

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