Download 6.1 B – Reciprocal, Quotient, and Pythagorean Identities

Name: ____________________
Pre- Calculus 12
Date: _____________
Chapter 6 – Trigonometric Identities
6.1 B – Reciprocal, Quotient, and Pythagorean Identities
Verifying and Proving Trigonometric Identities
Warm-up Exercise
Simplify by expressing each of the following in terms of a single trigonometric function.
Have your trigonometric identity sheet in front of you if necessary.
cos 
1.
sin 
cos 2 
2.
sin 2 
3. tan  sec cos 
4. 1  cos2 
5. cos 2  1
6. 1  tan2 
7. sec 2  1
8. 1  sec2 
9. 1  csc 2 
10. sin 2   cos2  1
11. csc 2   cot 2 
13.
1  sin 2 
1  tan 2 
14.
sec 2   1
csc 2   1
12. sin 2   cos2   tan2 
sin2   sin 
15.
cos   cos  sin 
Last day, you discovered that proving identities involved knowing how to work with rational
expressions. Today, you will learn a few more “tricks” involved when proving identities
involving the first 8 basic identities.
Complex Fractions
Identities often involve complex fractions. Simplify the following:
1 1

2 3
5
1
6
Use the technique you learned above to prove the following identities:
Example 1:
1  cos
 cos
1 sec 
Example 2:
1  tan
1  tan 

1  cot 
cot   1
Example 3:
sin   tan
 sin sec
1 cos 
Factoring
Recall how to factor a difference of squares a 2  b 2 
a trigonometric identity.
4
4
sin   cos  
Example 4:
Conjugate Trick
Recall how to rationalize a binomial denominator
Factoring can be used in the proof of
sin   cos 
2
2
2

1 3
This factor of 1  3 can be loosely called the “conjugate” . This technique can be used in some
special trigonometric identity proofs.
sin
1  cos
Example 5:

1  cos
sin
Common Denominators
Recall how to add two rational expressions with binomial denominators:
1
1


1  a 1 a
Obtaining a common denominator is often involved in an identity proof.
Example 6:
1
1

1  cos 1  cos

2csc 2 
6.1B Verifying and Proving Trigonometric Identities Assignment
Copy and prove the following identities:
1. sin  sec  csc    tan 1
2. sec 2  sec 2  sin 2   1
3. sin 4   cos4   2sin 2  1
4. tan 2  1  cot 2    sec 2 
5. cot  
1 cot 
1 tan
6.
1  sec 1  cos

sec   1 1  cos
sin   cos tan
 2tan  sin 
cot 
7.
1  sin csc   1

1  sin csc   1
8.
9.
sin
1  cos

1  cos
sin
10.
tan 
sec   1

sec   1
tan 
12.
2
2

 4sec 2 
1  sin 1  sin
14.
1  cos tan   sin 

sin
tan  sin 
11.
13.
sin
sin

 2csc 
1  cos 1  cos
1
1  sin

1  sin
cos2 