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Pre-Calculus
Unit 10 – Trigonometric Identities
Section 10.1
Simplifying
No Calculator
Simplify each of the following to a single trigonometric function or constant.
1. cos2 x  sin2 x
2.
1  cos x 1  cos x 
3.
4. 1  tan2 x
5.
 sec x  1sec x  1
6. tan2 x  sec 2 x
7. 1  cot 2 x
8.
csc x  1csc x  1
9.
10.
1
sin x
11. 1 
sin2 x
tan2 x
12.
 sin x  1 sin x  1
1
1

sin2 x tan2 x
1
1

2
cos x cot 2 x
13. cos x tan x
14. csc 2 x 1  cos2 x 
15. cos x  sec x  cos x 
16. cot x sec x sin x
17. cos2 x  sec 2 x  1
18. sin x csc x  sin x 
19. sin x tan x  cos x
20. csc x  cos x cot x
21.
 sec x  tan x sec x  tan x 
22.
1  cos x csc x  cot x 
23.
csc x  cot x sec x  1
24.
1  cos x 1  sec x  cos x
25.
sin x cos x
1  cos2 x
26.
tan x  cot x
sec 2 x
27.
 sin x  cos x 
28.
 sec
29.
cot 2 x
 sin x csc x
1  csc x
30.
tan2 x
1
sec x  1
32.
sec x  csc x
1  tan x
33.
cos x
1  sin x

1  sin x
cos x
2


x  1 csc 2 x  1
31. cos3 x  cos x sin2 x
2
  sin x  cos x 
2
Pre-Calculus
Unit 10 – Trigonometric Identities
Section 10.2
Verifying Trigonometric Identities
No Calculator
Verify each of the identities below. Remember, you may work with the left side (think of taking a derivative) and CANNOT
change the right side (which would be the simplified answer for your derivative).
1. cot 2 x  cos2 x  sin2 x  csc 2 x
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
sin x cos x

 sin x csc x
csc x sec x
tan2 x  sin2 x  tan2 x sin2 x
sin x
tan x

sin x  cos x 1  tan x
sin2 x
1
 cos x
1  cos x
cos x  sin x  sin3 x
 cot x  cos2 x
sin x
tan x  cot x  sec x csc x  0
1
1

 2 sec 2 x
1  sin x 1  sin x
 csc x  cot x  csc x  cot x   1
tan x  cot x
 sin2 x  cos2 x
tan x  cot x
cot x  tan x
 csc 2 x  sec 2 x
sin x cos x
1  sin2 x
 sin2 x cos2 x
1  cot 2 x
tan2 x
 sin2 x
2
1  tan x
tan x
1  sec x

 2 csc x
1  sec x
tan x
1  sin x 1  sin x

 4 tan x sec x
1  sin x 1  sin x
Pre-Calculus
Unit 10 – Trigonometric Identities
Section 10.3
Double Angle Identities
No Calculator
Verify each of the identities below. Remember, you may work with the left side (think of taking a derivative) and CANNOT
change the right side (which would be the simplified answer for your derivative).
1. cos4 x  sin4 x  cos 2x
2.
 4 sin x cos x  1  2 sin2 x   sin 4x
3.
1  cos 2x
 sin 2x
cot x
1  tan x  1  cos 2x   tan
2
4.
2
2
x
1  cos 2x
 tan2 x
1  cos 2x
6. 9 sec 2 x  5 tan2 x  5  4 sec 2 x
cos x
7. tan x 
 sec x
1  sin x
5.
8.
9.
10.
11.
12.
13.
14.
15.
 sin x  cos x    sin x  cos x 
cos x  tan x  cot x   csc x
2
2
tan x  cot x
1
sec x csc x
1
csc 2x  sec x csc x
2
2 tan x cos2 x  sin 2x
cos 2x  1
 sin x
2 sin x
sin x tan x  cos 2x sec x  cos x
1  tan2 x
 cos 2x
1  tan2 x
2