MTH 112 Elementary Functions Chapter 6 Trigonometric Identities, Inverse Functions, and Equations Section 3 Proving Trigonometric Identities

Review Identities Identities from Chapter 5 –Reciprocal relationships –Tangent & cotangent in terms of sine and cosine –Cofunction relationships –Even/odd functions Identities from Chapter 6, Sections 1 & 2 –Pythagorean –Sum & Difference –Cofunction relationships (including a few more than above) –Double Angle –Half Angle

Properties of Equality Reflexive  a = a Symmetric  If a = b then b = a. Transitive  If a = b and b = c, then a = c.

Proving Identities – Method 1 To prove that f(x) = g(x) … –Using known identities, change f(x) step by step until you get g(x). OR –Using known identities, change g(x) step by step until you get f(x).

Proving Identities – Method 2 To prove that f(x) = g(x) … –Using known identities, simplify/modify f(x) –Using known identities, simplify/modify g(x) –Continue the above two steps until you reach equivalent expressions.

Hints for Proving Identities There is not general rule to tell you what to do first! Know and recognize the identities. –This tells you what you can do. Work with the more complex side of the equation first. Use basic algebraic manipulations. With rational expressions, multiply by 1 (i.e. h(x)/h(x)). Change everything to sine and/or cosine. Combine or split fractions: (a+b)/c = a/c + b/c Multiply to remove parenthesis or factor. Try something. If it doesn’t work, try something else.