1. 9-1: Identities and Proofs

2. 9-1: Identities and Proofs
Option 1 (Graphing)
We did this back in 7-4
Example 1: Are either of the following equations identities?
2 sin2 x – cos x = 2 cos2 x + sin x
Identities will have the same graphs
Graphing examples on board
 Identity

3. 9-1: Identities and Proofs
Basic Trigonometric Identities
You should know these by now
Quotient Identities
Reciprocal Identities

4. 9-1: Identities and Proofs
Basic Trigonometric Identities (continued)
Periodicity Identities
sin(x ± 2π) = sin x csc(x ± 2π) = csc x
cos(x ± 2π) = cos x sec(x ± 2π) = sec x
tan(x ± π) = tan x cot(x ± π) = cot x
Pythagorean Identities
sin2 x + cos2 x = 1
tan2 x + 1 = sec2 x
1 + cot2 x = csc2 x
Negative Angle Identities (new! Though not really used…)
sin(-x) = -sin x cos(-x) = cos x tan(-x) = -tan x

5. 9-1: Identities and Proofs
Strategies to prove identities
Use algebra and previously proven identities (the ones you just saw) to transform one side of the equation into the other.
If possible, write the entire equation in terms of one trigonometric function.
Express everything in terms of sine and cosine.
Deal separately with each side of the equation A = B. First use identities and algebra to transform A into some expression C, then use (probably different) identities and algebra to transform B into the same expression C. Conclude that A = B.
Prove that AD = BC, with B ≠ 0 and D ≠ 0. You can then conclude that A/B = C/D.

6. 9-1: Identities and Proofs
Example 3: Transform One Side into the Other Side
Verify that
Reorder terms
Pythagorean Identity
Split the fraction ( )
and quotient identity

7. 9-1: Identities and Proofs
Example 4: Writing Everything in Terms of Sine/Cosine
Simplify (csc x + cot x)(1 – cos x)
(csc x + cot x)(1 – cos x)

8. 9-1: Identities and Proofs
Example 5A: Transform One Side into the Other Side
Prove that
Let’s work the left side
Multiply denominator by conjugate
FOIL denominator
Pythagorean identity
Divide sin x

9. 9-1: Identities and Proofs
Example 5B: Transform One Side into the Other Side
Prove that (alternate solution)
Still working the left side
Multiply each side by sin x
Multiply
Pythagorean identity
Difference of perfect squares
Divide by (1 + cos x)

10. 9-1: Identities and Proofs
Example 6: Dealing with Each Side Separately
Simplify
Working the left side – writing in terms of cos/sin
We just proved that in the last problem.
By the transitive property,

11. 9-1: Identities and Proofs
Example 7: Identities Involving Fractions
Prove the first identity, then use the first identity to prove the second identity
sec x(sec x – cos x) = tan2 x
sec2 x – 1 = tan2 x
tan2 x = tan2 x
Continue manipulating the equation by dividing by both denominators in (b)

12. 9-1: Identities and Proofs
Example 8: Using AD = BC, to prove A/B = C/D.
You can’t simply cross-multiply, because that would presume that the identity is true to begin with.
However, you can use the cross-multiplication plus some other method to prove the fractional identity true.
The idea is that if you prove some AD = BC, then you can conclude A/B = C/D.
Prove that (next slide)

13. 9-1: Identities and Proofs
Start with the cross-product
(cot x – 1)(1 + tan x) = (cot x + 1)(1 – tan x)
FOIL both sides
cot x + cot x tan x – 1 – tan x = cot x – cot x tan x + 1 – tan x
Use reciprocal identities
cot x + 1 – 1 – tan x = cot x – – tan x
cot x – tan x = cot x – tan x

14. 9-1: Identities and Proofs
From the book (pg 579), and is worth repeating:
“It takes a good deal of practice, as well as much trial and error, to become proficient at providing identities. The more practice you have, the easier it will become. Because there are many correct methods, your proofs may be quite different from those of your classmates, instructor, or text answers.”

15. 9-1: Identities and Proofs
Assignment
Page 580
Problems 5 – 8 (all) – Prove algebraically, not graphically
Problems 9 – 39 (odds)