Verifying Trig Identities

Overview

This lecture explains strategies for verifying trigonometric identities, focusing on algebraic manipulation and the use of fundamental trig identities through worked examples.

Introduction to Verifying Trig Identities

• Verifying trig identities means simplifying both sides of an equation to show they are equal.
• The process relies on known trig identities and algebraic steps to prove the equality.

Key Strategies for Verification

• Rewrite trig functions in terms of sine and cosine for easier manipulation.
• Use the Pythagorean identity: sin²x + cos²x = 1 to simplify expressions.
• Combine or split fractions as needed to make simplification possible.
• Multiply by conjugates to eliminate sums or differences in denominators.

Worked Examples

• csc x × tan x = sec x: Rewrite using definitions, cancel terms, and show both sides are equal.
• sin x tan x + cos x = sec x: Express all terms in sines and cosines, combine fractions, use identities, and simplify.
• 1 + tan²x = sec²x: Rewrite in terms of sine and cosine, find a common denominator, and use the Pythagorean identity.
• (1 + sin x)/cos x + cos x/(1 + sin x) = 2 sec x: Use conjugates to simplify denominators, combine fractions, and simplify.
• (sin²x + cos²x + cot²x)/(1 + tan²x) = cot²x: Replace sin²x + cos²x with 1, use identities, rewrite terms, and simplify.

General Tips and Advice

• Check if expressions can be rewritten in terms of sine and cosine.
• Use algebraic operations like combining or splitting fractions, factoring, and flipping denominators when dividing.
• There are often multiple valid approaches; consistent algebraic logic will lead to the solution.
• Practice is key to mastering these techniques.

Key Terms & Definitions

• Trig Identity: An equation involving trigonometric functions that is true for all values in the domain.
• Pythagorean Identity: sin²x + cos²x = 1.
• Secant (sec x): 1/cos x.
• Cosecant (csc x): 1/sin x.
• Tangent (tan x): sin x/cos x.
• Cotangent (cot x): cos x/sin x.
• Conjugate: An expression formed by changing the sign between two terms in a binomial.

Action Items / Next Steps

• Practice verifying trig identities with a range of problems.
• Review and memorize basic trig identities and definitions.
• Complete any assigned comprehension checks.