cnsaicnaincfai

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