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

Trig Identities IIdentity: an equation that is true for all values of the variable for which the expressions are defined EEx: or (x + 2) = x + 2 CConditional Equation: only true for some of the values EEx: tan x = 0 or x 2 + 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 + cot 2 x = csc 2 x