1. 7.1.1 Trig Identities and Uses

2. We have already discussed a few example of trig identities
All identities are meant to serve as examples of equality
Convert one expression into another
Can be used to verify relationships or simplify expressions in terms of a single trig function or similar

3. Past Identities Past identities we have already talked about include:
1) Reciprocal Identities
2) Quotient identities (tan, cot)
3) Cofunction identities

4. Period Identities
Recall, the period of a trig function is how often/over how many radians the values for the trig function repeat
sin(x + 2π) = sin(x)
cos(x + 2π) = cos(x)
tan(x + π) = tan(x)
csc(x + 2π) = csc(x)
sec(x + 2π) = sec(x)
cot(x + π) = cot(x)

5. Odd/Even
Looking at the values of particular functions on the unit circle reveals some other information
Odd: f(-x) = -f(x)
Opposite x yields opposite y-value
Even: f(-x) = f(x)
Opposite x yields the same y-value

6. Odd/Even sin(-x) = -sin(x) csc(-x) = -csc(x)
cos(-x) = cos(x) sec(-x) = sec(x)
tan(-x) = -tan(x) cot(-x) = -cot(x)

7. Pythagorean Identities
We can let x = cos(x) and y = sin(x) in terms of x/y coordinates on the coordinate grid
The same may apply to triangles

8. Our end goal is to with a simpler form of a long, entirely winded, expression
Most times, we will need to use more than one trig identity to help us

9. Example. Simplify the expression:
cos(x) + sin(x) tan(x)

10. Example. Simplify the expression:
sec(x) cot(x) – cot(x) cos(x)

11. Example. Simplify the expression:
sin2(x) – cos2(x)sin2(x)

12. Assignment
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