The document covers fundamental concepts of identities in trigonometry, providing examples of basic reciprocal, quotient and Pythagorean identities. It presents problems to practice identifying and using trigonometric identities, including filling in missing terms, simpl

2. What is an Identity?
An identity is a statement that two
expressions are equal for every value
of the variable.
Examples:
xxx 2
The left-hand expression always equals the
right-hand expression,
no matter what x equals.

3. The fundamental Identities
Reciprocal Identities Quotient Identities
x
x
x
Secx
x
x
tan
1
cot
cos
1
sin
1
csc



x
x
x
x
x
x
sin
cos
cot
cos
sin
tan



4. xx
xx
xx
22
22
22
csccot1
sec1tan
1cossin



Pythagorean Identities
X

5. A. Supply the missing term to complete
the following identities
1. Cos x Sec x = _______
2. Tan x . ______ = 1
3. 1 + _______ = Sec² x
4. Csc² x – 1 = _______
5. Cos x = Sin x / _______

6. 6. ______ = 1/Sin x
7. 1 – Cos² x= ______
8. Csc²x – ______ = 1
9. Cot x Sin x = _______
10. Sec²x– Tan² x= _______

7. B. Simplify the following:
1. Simplify Cosθ Cscθ Tanθ to 1.

8. 2. Simplify Cot²θ + 1__ into Cotθ
Cotθ Sec²θ

9. 3. Transform ( 1- Cosθ) ( 1 + Cosθ) into
Sin²θ

10. 4. Simplify Sinθ Secθ Cotθ into 1.

11. 5. Transform Sec θ – Sec θ Sin²θ into
Cosθ

12. Change everything on both sides to
sine and cosine.
Suggestions
• Start with the more complicated side
• Try substituting basic identities (changing all
functions to be in terms of sine and cosine may
make things easier)
• Try algebra: factor, multiply, add, simplify, split up
fractions
• If you’re really stuck make sure to:

13. Don’t Get Discouraged!
• Every identity is different
• Keep trying different approaches
• The more you practice, the easier it will be to
figure out efficient techniques
• If a solution eludes you at first, sleep on it!
Try again the next day. Don’t give up!
• You will succeed!

14. Simplify the following by
transforming into single term
1. Cscθ – Cscθ Cos²θ
2. (Sinθ + Cosθ)² + ( Sinθ – Cosθ)²
3. Sinθ Sec θ
4. Sin²θ Cot²θ Tan²θCsc²θ
5. Csc²θ ( 1 – Cos²θ)

15. 6. ( 1-sin²θ) ( 1 + Tan²θ)
7. SinA /CscA + CosA/SecA
8. 1 – Cos²A
1+SinA
9. Tan A Sin A + Cos A
10. Cot B + _SinB
1 + Cos B

16. 11. Sin² B Sec² B – Sec² B
12. Tan² θ - Tan² θ Sin²θ
13. Sinθ Secθ Cotθ
14. Sin²θ Sec²θ – Sec²θ
15. Secθ - SecθSin²θ

17. Prove that Sin A = Cos A Tan A
Statements Reasons
_____________ Given
_____________ Quotient Relationship
_____________ Cancellation

18. Prove that Csc A – Sin A = Cos A Cot A
Statements Reasons
_____________ Given
_____________ Reciprocal Identity
_____________ Addition of Fraction
_____________ Pythagorean Identity
_____________ Factoring
_____________ ______________

19. Proving Identities
1. Prove that Tan²x + 1 = Sec²x
2. Prove that 1 + Sec² x = Cos²x + 1
Sec²x
3. Verify Sec²x + Csc²x = Sec⁴ x
Cot²x

20. 4.Prove that Tan²x – Sin²x = Sin⁴ x Sec²x
5. Cotθ Sinθ Cosθ = 1 – Sin²θ
6. Tanθ + Cotθ = Csc θ Secθ
7. Cosθ Tanθ = Sec²θ - Tan²θ
Sinθ

21. 8. 1 + 2Cot²θ + Cot⁴θ = Csc ⁴θ
9. 2Sin²θ – 1 = Sin⁴θ – Cos⁴θ
10. 2Sec²θ = 1/1 +Sinθ + 1/ 1-Sinθ

22. C. Prove the ff: identities:
1. Tanx Sinx + Cosx = Secx
2. Secx – Tanx Sinx = Cosx
3. Tan²x = Secx – 1
1 + Secx
4. Csc²x – Cos²x = 1
Sin²x
5.Tan x + Cot x = Cscx/Cosx