Verifying Trig Identities (5.1)

Verifying Trig Identities(5.1) JMerrill, 2009 (contributions from DDillon)

Trig Identities • Identity: an equation that is true for all values of the variable for which the expressions are defined • Ex: or (x + 2) = x + 2 • Conditional Equation: only true for some of the values • Ex: tan x = 0 or x2 + 3x + 2 = 0

Recall

Recall - Identities Reciprocal Identities Also true:

Recall - Identities Quotient Identities

Fundamental Trigonometric Identities Negative Identities (even/odd) These are the only even functions!

Recall - Identities Cofunction Identities

Recall - Identities Pythagorean Identities

Simplifying Trig Expressions • Strategies • Change all functions to sine and cosine (or at least into the same function) • Substitute using Pythagorean Identities • Combine terms into a single fraction with a common denominator • Split up one term into 2 fractions • Multiply by a trig expression equal to 1 • Factor out a common factor

Simplifying # 1

Simplifying #2

Simplifying #3

Simplifying #4

Simplifying #5

Proof Strategies • Never cross over the equal sign (you cannot assume equality) • Transform the more complicated side of the identity into the simpler side. • Substitute using Pythagorean identities. • Look for opportunities to factor • Combine terms into a single fraction with a common denominator, or split up a single term into 2 different fractions • Multiply by a trig expression equal to 1. • Change all functions to sines and cosines, if the above ideas don’t work. ALWAYS TRY SOMETHING!!!

Example • Prove • 2 fractions that need to be added: • Shortcut:

1 + cot2x = csc2 x

Verifying Trig Identities (5.1)