Proving Trigonometric Identities

Prove complex trigonometric identities step-by-step. Grade 12 Math 30-1.

Lesson 5.6 of Trigonometric Identities in Math 30-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• Equations vs Identities & Counterexamples. Understand the difference between a trigonometric equation and a trigonometric identity, learn the key strategies for proving identities, and practice finding counterexamples to disprove false claims.
• Proving Identities Part 1. Simplify trig expressions to single ratios, algebraically prove identities while stating all NPVs, and determine unknown expressions within identities.
• Proving Identities Part 2. Use reciprocal, Pythagorean, sum difference, and double-angle identities together to prove more complex trigonometric identities.

Inside the lesson — a free preview

Equation vs Identity

These two words look similar but mean very different things in math. It's important to understand the difference before we start proving identities.

What is a Trigonometric Equation?. A trigonometric equation is a statement that is only true for certain specific values of the variable.

Example:

θ + 1 = (3)/(2) where 0 ≤ θ ≤ 2π

This is only satisfied when θ = (π)/(3) or θ = (5π)/(3). For any other value of θ, the equation is FALSE.

What is a Trigonometric Identity?. A trigonometric identity is a statement that is true for ALL values of the variable (except where any denominator equals zero).

Example:

1 - ^2θ = θθθ

No matter what angle θ you substitute (where θ ≠ 0), the left side will always equal the right side. It's true for every valid angle — that's what makes it an "identity".

Strategies for Proving Trigonometric Identities

Proving a trig identity means showing that the left side (LS) simplifies to become identical to the right side (RS). Tap each strategy below to expand it.

Getting Started

Now that we have our strategies, we can start proving identities algebraically. Remember to always state NPVs and only work on one side at a time.

Worked example: Counterexamples

A counterexample is a specific example that proves a claim is false.

While no amount of examples can fully prove an identity is always true, it only takes one counterexample to disprove it. If you can find even one value of the variable where the two sides are NOT equal, the statement is not an identity.

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