1. Trigonometric Identitiesor
“Algebra dresses Trig up to make it super sexy for Calculus”

2. Two functions f and g are said to be identically equal if
for every value of x for which both functions are defined. Such an equation is referred to as an identity. An equation that is not an identity is called a conditional equation.

3. Reciprocal Identities
Quotient Identities

4. Periodic Properties
These can be rewritten with the product of degrees and a constant k as …

5. Even-Odd Properties

6. Pythagoras gets involved …
Trigonometric identities
The fundamental identity
The fundamental trigonometric identity is derived from Pythagoras’ theorem

7. Identities using the trig forms of Pythagoras
Use this form to get the rest 1
sin 
cos
Important in integration
Less important

8. Trigonometric identities
Two more identities
Dividing the fundamental identity by cos2
Dividing the fundamental identity by sin2

9. Looking Ahead to Calculus
Work with identities to simplify an expression with the appropriate trigonometric substitutions.
For example, if x = 3 tan , then

19. Prove
Notice the LHS has a mixture of cos and sin terms where the RHS only has cos terms
Strategy: Turn sin into cos using sin2 + cos2 = 1

20. Consider the LHS
Turn sin into cos using sin2 + cos2 = 1

21. LHS

22. LHS
Expand bracket

23. LHS

24. LHS
Collecting terms
=RHS

25. 3 x 5

26. Trigonometric Formulas
Sums and differences of angles

27. Example 1
Expand
Strategy: Use the formula for cos (A-B)

28. Expand
Sketch the trig graphs to get the values

29. Expand

30. Expand

31. Example 2
Prove
LHS
Strategy: You know the addition formulae for sin and cos, so replace tan by sin/cos

32. Prove
LHS
Strategy: Use the addition formulae to expand sin and cos

33. Prove
LHS
Strategy: Turn the sin and cos terms back into tan by dividing evert term in both numerator and denominator by coscos

34. Prove
LHS
Strategy: Turn the sin/cos terms back into tan and cancel cos/cos terms.

35. Prove
=RHS

36. Trigonometric Formulas
Double angles

37. Trigonometric Formulas
Half angles

38. Example 3
Prove
Strategy: The RHS contains no double angles so expand the LHS double angles.
Question: Which of the 3 forms to use for cos?
Answer: As the RHS, being cot, must have sin in the denominator, use the form involving sin

39. Prove
Simplify and cancel

40. Prove

41. Prove

42. Trigonometric Formulas
Sums and differences of ratios

43. Trigonometric Formulas
Products of ratios

44. Guidelines for Establishing Identities
1. It is almost always preferable to start with the side containing the more complicated expression.
2. Rewrite sums or differences of quotients as a single quotient.
3. Sometimes rewriting one side in terms of sines and cosines only will help.
4. Always keep your goal in mind.