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

RECIPROCAL IDENTITIES QUOTIENT IDENTITIES PYTHAGOREAN IDENTITIES

The first type of examples are rewriting expressions (no = sign) Rewrite in terms of a single trig function

Rewrite in terms of cos

Rewrite in terms of sin

Another type of example is verifying an identity Verify You will use identities you already to know to prove an equation that you are given

Verify 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

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

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! Hints for Establishing Identities