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. We are done! We've shown the LHS equals the RHS
Verify 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 until both sides match.

4. Verify the following identity:
Let's sub in here using reciprocal identity and quotient identity
combine fractions
Another trick if the you have 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
FOIL numerator

5. 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 

6. 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