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**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.

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**RECIPROCAL IDENTITIES**

QUOTIENT IDENTITIES PYTHAGOREAN IDENTITIES EVEN-ODD IDENTITIES

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**We are done! We've shown the LHS equals the RHS**

Establish the following identity: Let's sub in here using reciprocal identity We are done! We've shown the LHS equals the RHS We often use the Pythagorean Identities solved for either sin2 or cos2. sin2 + cos2 = 1 solved for sin2 is sin2 = 1 - cos2 which is our left-hand side so we can substitute. 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.

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**We worked on LHS and then RHS but never moved things across the = sign**

Establish the following identity: Let's sub in here using reciprocal identity and quotient identity We worked on LHS and then RHS but never moved things across the = sign FOIL denominator combine fractions 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

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**Hints for Establishing Identities**

Get common denominators If you have squared functions look for Pythagorean Identities Work on the more complex side first If you have a denominator of 1 + trig function try multiplying top & bottom by conjugate and use Pythagorean Identity When all else fails write everything in terms of sines and cosines using reciprocal and quotient identities Have fun with these---it's like a puzzle, can you use identities and algebra to get them to match! MathXTC

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

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