1. 7.2 Verifying Trigonometric Identities
By the end of the period students will use the basic trig identities to verify other trig identities, as evidenced by completion of a collaborative poster.

2. Proofs of trigonometric Identities
By the end of the period students will use the basic trig identities to verify other trig identities, as evidenced by completion of a collaborative poster.
Proofs of trigonometric Identities
Transform the more complicated side into the more simple side. (This is what we did in example 3 for 7.1)
You are ONLY allowed to work on one side of the equation.
Use the basic trig functions, sin 𝑥 and cos 𝑥 , as much as possible.
Factor or multiply to simplify.
Use 𝑎+𝑏 𝑎−𝑏 = 𝑎 2 − 𝑏 2
Multiply by the equivalent of cos 𝑥 1+ cos 𝑥
Add 0, − sin 𝑥 + sin 𝑥 =0
Watch for Pythagorean identities… all of them

3. Pythagorean Identities
By the end of the period students will use the basic trig identities to verify other trig identities, as evidenced by completion of a collaborative poster.
Pythagorean Identities
sin 2 𝑥 + cos 2 𝑥 =1
sin 2 𝑥 −1=− cos 2 𝑥
cos 2 𝑥 −1=− sin 2 𝑥
1− sin 2 𝑥 = cos 2 𝑥
1− cos 2 𝑥 = sin 2 𝑥
1+ cot 2 𝑥 = csc 2 𝑥
1= csc 2 𝑥 − cot 2 𝑥
1− csc 2 𝑥 =− cot 2 𝑥 …
tan 2 𝑥 +1= sec 2 𝑥
All of the Pythagorean identities have a square in them…look for that!

4. Example 1: Verify the trig identity
By the end of the period students will use the basic trig identities to verify other trig identities, as evidenced by completion of a collaborative poster.
Example 1: Verify the trig identity
Steps
Justifications A.
tan 2 𝑥 = sec 2 𝑥 − tan 𝑥 ∙ cot 𝑥 = sec 2 𝑥 − tan 𝑥 ∙ 1 tan 𝑥 = sec 2 𝑥 −1 = tan 2 𝑥
Given
Reciprocal ID
Reduce
Pythagorean identity

5. Example 1: Verify the trig identity
By the end of the period students will use the basic trig identities to verify other trig identities, as evidenced by completion of a collaborative poster.
Example 1: Verify the trig identity
Steps
Justifications B.
1= cos 2 𝑥 ∙ tan 2 𝑥 + cos 2 𝑥 = cos 2 𝑥 tan 2 𝑥 +1 = cos 2 𝑥 sec 2 𝑥 = cos 2 𝑥 ∙ 1 cos 2 𝑥 =1
Given
Common factor
Pythagorean identity
Reciprocal identity
reduce

6. Example 1: Verify the trig identity
By the end of the period students will use the basic trig identities to verify other trig identities, as evidenced by completion of a collaborative poster.
Example 1: Verify the trig identity
Steps
Justifications C.
tan 𝑥 = 1+ tan 2 𝑥 csc 𝑥 ∙ sec 𝑥 = sec 2 𝑥 csc 𝑥 ∙ sec 𝑥 = sec 𝑥 ∙ sec 𝑥 csc 𝑥 ∙ sec 𝑥 = sec 𝑥 csc 𝑥 = sec 𝑥 1 ∙ 1 csc 𝑥 = 1 cos 𝑥 ∙ sin 𝑥 1 = tan 𝑥
Given
Pythagorean identity
Reduce num/den
Rewrite
Reciprocal identities
Quotient identity

7. Example 1: Verify the trig identity
By the end of the period students will use the basic trig identities to verify other trig identities, as evidenced by completion of a collaborative poster.
Example 1: Verify the trig identity
Steps
Justifications D.
7 sec 𝜃 +5 csc 𝜃 = 7 sin 𝜃 +5 cos 𝜃 sin 𝜃 cos 𝜃 = 7 sin 𝜃 sin 𝜃 cos 𝜃 + 5 cos 𝜃 sin 𝜃 cos 𝜃 = 7 cos 𝜃 + 5 sin 𝜃 =7 sec 𝜃 +5 csc 𝜃
Given
𝑎+𝑏 𝑝 = 𝑎 𝑝 + 𝑏 𝑝
Reduce num/den
Reciprocal property

8. Example 1: Verify the trig identity
Steps
Justifications E.
csc 2 𝐴 − cot 2 𝐴 = sin 𝐴 csc 𝐴 + cos 𝐴 sec 𝐴
= sin 𝐴 ∙ sin 𝐴 + cos 𝐴 ∙ cos 𝐴
= sin 2 𝐴 + cos 2 𝐴 =1
??? WHAT????
We want cot 2 𝐴 , so lets get it
=1+ cot 2 𝐴 − cot 2 𝐴
= csc 2 𝐴 − cot 2 𝐴
Given
Reciprocal property
Simplify
Pythagorean identity
Add 0

9. Example 1: Verify the trig identity
By the end of the period students will use the basic trig identities to verify other trig identities, as evidenced by completion of a collaborative poster.
Example 1: Verify the trig identity
Steps
Justifications
cos 𝑥 = sec 𝑥 − tan 𝑥 1+ sin 𝑥
= sec 𝑥 + sec 𝑥 ∙ sin 𝑥 − tan 𝑥 − tan 𝑥 sin 𝑥
= 1 cos 𝑥 cos 𝑥 ∙ sin 𝑥 − sin 𝑥 cos 𝑥 ? − sin 𝑥 cos 𝑥 ∙ sin 𝑥
= 1 cos 𝑥 − sin 2 𝑥 cos 𝑥
= 1− sin 2 𝑥 cos 𝑥
= cos 2 𝑥 cos 𝑥
= cos 𝑥 F.
Given
Distributive property (FOIL, etc)
Reciprocal property
Quotient property
Sum to 0
Simplify
Combine fractions
Pythagorean ID
Reduce num/den

10. Example 1: Verify the trig identity
By the end of the period students will use the basic trig identities to verify other trig identities, as evidenced by completion of a collaborative poster.
Example 1: Verify the trig identity
Steps
Justifications G.
tan 2 𝜃 − sin 2 𝜃 = tan 2 𝜃 ∙ sin 2 𝜃 = sec 2 𝜃 −1 ∙ sin 2 𝜃 = sec 2 𝜃 ∙ sin 2 𝜃 − sin 2 𝜃 = 1 cos 2 𝜃 ∙ sin 2 𝜃 − sin 2 𝜃 = tan 2 𝜃 − sin 2 𝜃
Given
Pythagorean identity
Distributive property
Reciprocal identity
Quotient identity

11. Summary What is the next step in the proof of
By the end of the period students will use the basic trig identities to verify other trig identities, as evidenced by completion of a collaborative poster.
Summary
What is the next step in the proof of
sin 2 𝑥 = sec 2 𝑥 −1 sec 2 𝑥
sin 2 𝑥 = tan 2 𝑥 sec 2 𝑥
sin 2 𝑥 = ∙
Using the fact that sin 2 𝜃 + cos 2 𝜃 =1, show that
cos 2 𝜃 − sin 2 𝜃 =2 cos 2 𝜃 −1
*hint: how can I get 2 cos 2 𝜃 ’s on the left side?

12. Summary What is the next step in the proof of
By the end of the period students will use the basic trig identities to verify other trig identities, as evidenced by completion of a collaborative poster.
Summary
What is the next step in the proof of
sin 2 𝑥 = sec 2 𝑥 −1 sec 2 𝑥
sin 2 𝑥 = tan 2 𝑥 sec 2 𝑥
sin 2 𝑥 = sin 2 𝑥 cos 2 𝑥 ∙ 1 sec 2 𝑥
Using the fact that sin 2 𝜃 + cos 2 𝜃 =1, show that
cos 2 𝜃 − sin 2 𝜃 =2 cos 2 𝜃 −1
*hint: how can I get 2 cos 2 𝜃 ’s on the left side?
cos 2 𝜃 + cos 2 𝜃 − sin 2 𝜃 − cos 2 𝜃 =2 cos 2 𝜃 −1
2 cos 2 𝜃 − sin 2 𝜃 + cos 2 𝜃 =2 cos 2 𝜃 −1
2 cos 2 𝜃 −1=2 cos 2 𝜃 −1

13. Partner Poster
As a pair, put the steps of your trigonometric proof in order.
For each step determine which basic trig identity was used and write that in the justification side of the proof
Create a poster to display your work.
NEAT and Legible
Use of color for clarity