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Analytic Trig Test REVIEW
Using Trig Identities
Name ____________________________________
Use the fundamental trigonometric identities to
determine the simplified form of the expression.
1. cot ßsin ß
2. sin ßsec ß
3.
tan ß
sin ß
10. If tan  
9
, find cot  .
40
Factor the expression and use the fundamental
identities to simplify.
11. cos2 xsec2 x  cos2 x
12. cos2 xsin2 x  cos2 x
4. cot ßtan ß
Which expression completes the fundamental
trigonometric identity?
5. cos  x 
bg
6. cot b
 xg

7. csc b
 xg

8. Is the given a trig identity?
1
[A] secu
cosu

[B] sin u sinu
2
2
[C] tan ucot 2 u1
1
[D] secusin
u
2
[E] tan ucot 2 u1

[F] sin u sinu
2
1
[G] secu
cosu
sinu
[H] cotu
cosu
Let  be an acute angle. Use the given function
value and trigonometric identities to find the
indicated trigonometric function.
8
9. If sin   , find csc  .
17
F
IJ
G
H K
F
I
G
H J
K
Convert all of the terms to sines and cosines and
simplify to find the expression that completes the
identity.
tan x
13.

sec x
14.
secx

tanxcscx
15.
tan x

csc x
Add or subtract the fractions. Then simplify using
the Pythagorean identities and factoring to find the
expression that completes the identity.
1  sinx
cosx
16.

cosx
1  sinx
17.
sinx
cosx  1

cosx  1
sinx
18.
cot x
csc x  1

cscx  1
cotx
Verify the identity.
cscx
csc x
19.

 2 secx
secx  tanx secx  tan x
20.
sin x
sec x tan x
1sin 2 x
Show that the equation is an identity.
1secu
tanu
21.

 2cotu
tanu
1secu
22.
1cosu
sinu

 2cscu
sinu
1cosu
23.
cos x
1  sinx

1  sin x
cos x
Identify the x-values that are solutions of the
equation.
24. 5 3tanx  3  8 3tanx
25. 3tanx  2  5tanx
Find the expression that completes the identity.

F
I
 xJ
G
H4 K
3
I
37. sin F
 xJ
G
H4 K
36. cos
38. Find the exact value of the expression.

11

11
cos cos
 sin sin
12
12
12
12
39. Find the exact value of cos 2x using the double
angle formula.
2 
cos x   ,
 x
13 2
40. Find the exact solutions to the equation in the
interval 0, 2 . 2sin 4 x  2cos 2 x
26. 9tanx  8 3  17tanx
g
27. 2tanx  7 3  9tanx
Use the figures to
find the exact value of the
trigonometric function.
41.
24
tan 2
28. 3cot 2 x  9  0
Find all solutions of the equation in the
interval 0, 2 .
g

45
29. 2cosxsinx sinx 0
30. tan 2   
6
3
sec 
6

2

8
31. 2cot 2 x3 cscx  0
32. 2csc3x 
42. cot
Use the power-reducing formulas to find the exact
value of the trigonometric function.
4
30
3
42. sin 2 15
x 5
33. 5sin 
30
2 2
x
x
34. 2 sec 2  3 sec  2  0
2
2
43. Express sin9 x  sin7 x as a product containing
only sines and/or cosines.
b g
35. Find the exact value of cos 285 .