Pythagorean identities

What You'll Learn

Derive the three Pythagorean identities from the unit circle and Pythagorean theorem

Apply sine²θ + cos²θ = 1 to prove tan²θ + 1 = sec²θ and 1 + cot²θ = csc²θ

Recognize when to substitute Pythagorean identities in trigonometric expressions

Use quotient and reciprocal identities to convert between trig functions

Simplify complex expressions by combining fractions and applying conjugates

What You'll Practice

Proving Pythagorean identities using the unit circle definition

Simplifying expressions with sec²x - 1 and other Pythagorean forms

Combining trigonometric fractions with common denominators

Multiplying by conjugates to create difference of squares patterns

Verifying complex trigonometric identities step-by-step

Why This Matters

Pythagorean identities are essential tools you'll use throughout trigonometry, precalculus, and calculus. They allow you to simplify complex expressions, solve equations, and verify identitiesskills critical for understanding wave functions, circular motion, and advanced calculus techniques like integration.

Before You Start — Make Sure You Can:

Use sine ratio to calculate angles and sides (Sin = \( \frac{o}{h}\) )Use cosine ratio to calculate angles and sides (Cos = \( \frac{a}{h}\) )Factoring difference of squares: x^2 - y^2Unit circle

This Unit Includes

5 Video lessons

Practice exercises

Learning resources

Skills

Pythagorean Identities

Unit Circle

Trigonometry

Identity Proofs

Algebraic Manipulation

Conjugates

Simplification