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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