1. Objective
Use fundamental trigonometric identities to simplify and rewrite expressions and to verify other identities.

2. Pythagorean Theorem
x2 + y2 = r2
Note: This will also be the equation for the unit circle

4. A derivation for a Pythagorean identity is shown below.
x2 + y2 = r2
Pythagorean Theorem
Divide both sides by r2.
Substitute
cos θ for and sin θ for
cos2 θ + sin2 θ = 1

5. y r x

6. Reciprocal Identities

7. Quotient Identities y x = x y =

8. Do you remember the Unit Circle?
What is the equation for the unit circle?
x2 + y2 = 1
What does x = ? What does y = ?
(in terms of trig functions)
sin2θ + cos2θ = 1
1st Pythagorean Identity!

9. Take the Pythagorean Identity and discover a new one!
Hint: Try dividing everything by cos2θ
sin2θ + cos2θ = 1
sin2θ + cos2θ =
cos2θ cos2θ cos2θ
tan2θ = sec2θ
Quotient
Identity
Reciprocal
Identity
2nd
Pythagorean Identity

10. Take the Pythagorean Identity and discover a new one!
Hint: Try dividing everything by sin2θ
sin2θ + cos2θ = 1
sin2θ + cos2θ =
sin2θ sin2θ sin2θ
cot2θ = csc2θ
Quotient
Identity
Reciprocal
Identity
3rd
Pythagorean Identity

11. Opposite Angle Identities
sometimes these are called even/odd identities

12. Other versions of the Pythagorean identities may be used also.
Look at and solve for cos2 θ
cos2 θ + sin2 θ = 1
This should leave you with:
cos2 θ = 1 - sin2 θ
Do this again this time solving for sin2 θ.
This should leave you with:
sin2 θ = 1 - cos2 θ

13. If cot θ = 2 and cos θ < 0, find sin θ and cos θ.
Use Pythagorean Identities
If cot θ = 2 and cos θ < 0, find sin θ and cos θ.
Use the Pythagorean Identity that involves cot θ.
cot 2 θ + 1 = csc 2 θ Pythagorean Identity
(2) = csc 2 θ cot θ = 2
5 = csc 2 θ Simplify.
= csc θ Take the square root of each side.
Reciprocal Identity
Solve for sin θ.

14. You may start with either side of the given equation
You may start with either side of the given equation. It is often easier to begin with the more complicated side and simplify it to match the simpler side.
Helpful Hint
If you get stuck, try converting all of the trigonometric functions to sine and cosine functions.

15. Other strategies you may need to use are:
Get a common denominator.
Split up a fraction that has a monomial in the denominator.
Distribute.
Simplify complex fractions.