Introduction
You now know the Pythagorean Identities. But how do you prove that a complex expression like is equal to ? That's the skill we're building today.
Prerequisite Connection
You know the Pythagorean Identities and basic algebraic factoring techniques.
Today's Increment
We're learning systematic proof strategies: working one side, converting to sine/cosine, and factoring.
Why This Matters
Proving identities is a core skill in Calculus and Differential Equations, where simplifying before integrating can save hours of work.
Proof Strategies
Work One Side Only
Pick the more complicated side and transform it into the simpler side. Never cross the equals sign.
Convert to Sine and Cosine
When stuck, rewrite everything in terms of and . This often reveals simplifications.
Factor and Combine Fractions
Use algebraic techniques: factor out common terms, find common denominators, and use conjugates.
Worked Examples
Example 1: Basic Substitution
Verify:
Work the Left Side
Recognize from Identity #1.
Substitute
Simplify
✓
LHS = RHS. Identity verified. ✓
Example 2: Convert to Sine/Cosine
Verify:
Convert LHS
Find Common Denominator
Apply Identity #1
✓
LHS = RHS. Identity verified. ✓
Example 3: Factoring (Advanced)
Verify:
Find Common Denominator
Expand Numerator
Cancel and Simplify
✓
LHS = RHS. Identity verified. ✓
Common Pitfalls
Working both sides like an equation
You're proving it's an identity, not solving an equation. Only manipulate one side at a time.
Giving up too soon
If you're stuck, try converting everything to and . It almost always works.
Real-World Application
Computer Graphics & Animation
Game engines and 3D rendering software constantly simplify trigonometric expressions for efficiency. When rotating objects, reducing complex trig formulas to simpler forms using identities saves computational power, allowing for smoother animations and faster frame rates.
Practice Quiz
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