This document provides guidance on establishing trigonometric identities by substituting known identities and simplifying expressions until both sides of the equality are equivalent. It reviews basic reciprocal, quotient, and Pythagorean identities that can be used. Tips are given such as working with the more complex side first, multiplying the numerator and denominator by conjugates, and writing all expressions in terms of sines and cosines if needed. The goal is to use identities and algebra to prove two expressions are equal without moving terms across the equal sign.

1. TRIGONOMETRIC IDENTITIES Remember an identity is an equation that is true for all defined values of a variable. We are going to use the identities that we have already established to "prove" or establish other identities. Let's summarize the basic identities we have.

2. RECIPROCAL IDENTITIES QUOTIENT IDENTITIES PYTHAGOREAN IDENTITIES EVEN-ODD IDENTITIES

3. Establish the following identity: In establishing an identity you should NOT move things from one side of the equal sign to the other. Instead substitute using identities you know and simplifying on one side or the other side or both until both sides match. Let's sub in here using reciprocal identity We often use the Pythagorean Identities solved for either sin 2  or cos 2  . sin 2  + cos 2  = 1 solved for sin 2  is sin 2  = 1 - cos 2  which is our left-hand side so we can substitute. We are done! We've shown the LHS equals the RHS

4. Establish the following identity: Let's sub in here using reciprocal identity and quotient identity Another trick if the denominator is two terms with one term a 1 and the other a sine or cosine, multiply top and bottom of the fraction by the conjugate and then you'll be able to use the Pythagorean Identity on the bottom We worked on LHS and then RHS but never moved things across the = sign combine fractions FOIL denominator

6. Acknowledgement I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint. www.slcc.edu Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum. Stephen Corcoran Head of Mathematics St Stephen’s School – Carramar www.ststephens.wa.edu.au