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4.
THE FUNDAMENTAL TRIGONOMETRIC
IDENTITIES
A trigonometric equation is, by definition, an equation that involves at least one trigonometric
function of a variable. Such an equation is called a trigonometric identity if it is true for all
values of the variable for which both sides of the equation are defined. An equation that is not an
identity is called a conditional equation. For instance, the trigonometric equation
1
csc t =
sin t
is an identity, since it is true for all values of t ( except, of course, for those values of t for which
csct or 1 sin t is undefined). On the other hand, the trigonometric equation
sin t = cos t
is a conditional equation, since there are values of t (for instance, t = 0) for which it isn’t true.
y
Figure 4.1
unit circle
P=(x,y)
= (cos θ , sin θ )
θ
O
-θ
x
Q = ( x , –y )
= (cos(− θ ), sin(−θ ))
Now we are going to derive the
trigonometric identities
and
cos(– θ ) = cos θ .
sin(– θ ) = – sin θ
Figure 4.1 shows the angle θ and the corresponding angle – θ both in standard position.
Evidently, the points P and Q , where the terminal sides of these angles intersect the unit circle,
are mirror images of each other across the x axis. Therefore, if P = ( x , y ) then it follows that
Q = ( x , –y ). In section 2, we showed that
P = ( x , y ) = (cos θ , sin θ ).
Likewise,
Q = ( x , –y ) = (cos(– θ ) , sin(– θ ) ).
Therefore,
sin(– θ ) = –y = – sin θ
and
cos(– θ ) = x = cos θ .
If we now combine the identities above with the quotient identity, tan θ = sin θ cos θ , we find
that
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tan(– θ ) =
sin(- θ)
sinθ
=–
= – tan θ .
cos(- θ )
cos θ
Similar arguments apply to cot(– θ ), sec(– θ ), and csc(– θ ). The results are summarized in the
following theorem.
Even-Odd Identities
For all values of θ in the domains of the functions:
(i) sin(– θ ) = – sin θ (ii) cos(– θ ) = cos θ (iii) tan(– θ ) = – tan θ
(iv) cot(– θ )= – cot θ (ii) sec(– θ ) = sec θ (iii) csc(– θ ) = – csc θ
Notice that only the cosine and its reciprocal the secant are even functions – the remaining four
trigonometric functions are odd. The even – odd identities are often used to simplify expressions,
as in the following example:
Example 4.1 ---------------------------- -----------------------------------------------------------Use the even – odd identities to simplify each expression.
sin(−θ ) + cos(−θ )
(a)
(b) 1 + tan 2 (−t )
sin(−θ ) − cos(−θ )
(a)
sin(−θ ) + cos(−θ ) = − sin θ + cos θ = − (sin θ − cos θ ) = sin θ − cos θ
sin θ + cos θ
sin(−θ ) − cos(−θ ) − sin θ − cos θ
− (sin θ + cos θ )
(b) 1 + tan 2 (−t ) = 1 + [tan(−t )]2 = 1 + (− tan t ) 2 = 1 + tan 2 t = sec 2 t .
_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Fundamental Trigonometric Identities
1. csc θ =
1
sin θ
2. sec θ =
1
cos θ
3. cot θ =
1
tan θ
4. tan θ =
sin θ
cos θ
5. cot θ =
cos θ
sin θ
6. cos 2 θ + sin 2 θ = 1
7. 1 + tan 2 θ = sec 2 θ
8. 1 + cot 2 θ = csc 2 θ
9. sin(– θ ) = –sin θ
10. cos(– θ ) = cos θ
11. tan(– θ ) = –tan θ
12. cot(– θ ) = –cot θ
13. sec(– θ ) = sec θ
14. csc(– θ ) = –csc θ
Not only should you memorize these fourteen fundamental identities, but they should become so
familiar to you that you can recognize them quickly even when they are written in equivalent
forms. For instance, csc θ = 1 sin t can also be written as
1
(sin θ )(csc θ ) = 1
or
sin θ =
.
csc θ
Incidentally, a product of values of trigonometric functions such as (sin θ )(csc θ ) is usually
written simply as sin θ csc θ , unless the parentheses are necessary to prevent confusion.
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Example 4.2 ---------------------------- -----------------------------------------------------------Simplify each trigonometric expression by using the fundamental identities.
(a) (csc θ )(cos θ )
1
(csc θ )(cos θ ) =
cos θ
cos θ =
sin θ
sin θ
= cot θ
(b) tan 2 t − sec2 t
Because 1 + tan θ = sec θ , it follows that tan θ − sec θ = − 1 .
2
2
2
2
(c) csc4 x − 2 csc2 x cot 2 x + cot 4 x
2
2
2
2
The given expression is the square of csc x − cot x . Because cot x + 1 = csc x , we have
2
2
4
2
2
4
2
2
2
2
csc x − cot x = 1. Therefore, csc x − 2 csc x cot x + cot x = (csc x − cot x ) = 1 = 1.
_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
The reciprocal and quotient identities enable us to write csc θ , sec θ , tan θ , and cot θ in terms of
sin θ and cos θ . Therefore:
Any trigonometric expression can be written in terms of sines and cosines.
This fact and the Pythagorean identity cos 2 θ + sin 2 θ = 1 can often be used to simplify
trigonometric expressions.
Example 4.3 ---------------------------- -----------------------------------------------------------Rewrite each trigonometric expression in terms of sines and cosines, and then simplify the result.
(a) csc t –
cos t
sin t
csc t –
cot t
sec t
=
=
(b)
1
sin t
1
sin t
–
⎛ cos t ⎞
⎜
⎟
⎝ sin t ⎠
⎛ 1 ⎞
⎜
⎟
⎝ cos t ⎠
=
2
cos t
–
sin t
=
1
–
sin t
1 − cos 2 t
sin t
⎛ cos t ⎞ ⋅ cos t
⎜
⎟
⎝ sin t ⎠
2
= sin t = sin t
sin t
csc 2 x sec 2 x
csc 2 x + sec 2 x
⎛ 1 ⎞⎛ 1 ⎞
⎜ 2 ⎟⎜ 2 ⎟
csc x sec x
⎝ sin x ⎠⎝ cos x ⎠ =
=
2
2
csc x + sec x
⎛ 1 ⎞ ⎛ 1 ⎞
⎜ 2 ⎟+⎜ 2 ⎟
⎝ sin x ⎠ ⎝ cos x ⎠
2
2
=
1
2
2
cos x + sin x
=
1
⎛ 1 ⎞⎛ 1 ⎞
⎟⎜
⎟
⎝ sin 2 x ⎠⎝ cos 2 x ⎠
⎛ 1
1 ⎞
2
2
⎟
sin x ⋅ cos x ⋅ ⎜⎜ 2 +
⎟
⎝ sin x cos 2 x ⎠
2
2
sin x ⋅ cos x ⋅ ⎜
= 1.
1
_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
The Pythagorean identity cos 2 θ + sin 2 θ = 1 can be written as
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cos 2 θ = 1 – sin 2 θ
sin 2 θ = 1 – cos 2 θ .
or
Therefore, we have
( i ) sin θ = ± 1− cos 2 θ
( ii ) cos θ = ± 1− sin 2 θ
In either case, the correct algebraic sign is determined by the quadrant or coordinate axis
containing the terminal side of the angle θ in standard position. After you have rewritten a
trigonometric expression in terms of sines and cosines, you can use these equations to bring the
expression into a form involving only the sine or only the cosine.
Example 4.4 ---------------------------- ------------------------------------------------------------
Rewrite the expression cot θ csc 2 θ in terms of sin θ only.
± 1 − sin θ
cos θ
⎛ cosθ ⎞ ⎛ 1 ⎞
2
cot θ csc θ = ⎜
=
.
⎟. ⎜ 2 ⎟ =
3
3
sin θ
⎝ sin θ ⎠ ⎝ sin θ ⎠
sin θ
2
_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
Algebraic expressions not originally containing trigonometric functions can often be simplified
by substituting trigonometric expressions for the variable. This technique, called trigonometric
substitution, is routinely used in calculus to rewrite radical expressions as trigonometric
expressions containing no radicals.
Example 4.5 ---------------------------- ------------------------------------------------------------
If a is a positive constant, rewrite the radical expression a 2 − u 2 as a trigonometric expression
containing no radical by using the trigonometric substitution u = a sin θ .
Assume that – π < θ < π , so that cos θ > 0.
2
a2 − u2
2
= a − (a sin θ ) 2 = a 2 − a 2 sin 2 θ
2
= a 2 (1 − sin 2 θ ) = a 2 cos 2 θ
= a cos θ .
Section 4 Problems---------------------- -----------------------------------------------------------In problems 1 to 6, use the even-odd identities to simplify each expression.
1. sin (– θ ) cos (– θ )
3. tan ( t ) + tan ( – t )
1 + csc (−α )
5. 1 − cot (− β )
2. cot 2 (−u ) + 1
4. cos( –x ) sec x
6. [ 1 + sin γ ][ 1 + sin(– γ )]
In problems 7 to 28, use the fundamental identities to simplify each expression.
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7. secθ ⋅ sinθ
8. 1 +
9. cot υ ⋅ sec υ
10.
csc β
sec β
12.
11.
13. cot 2 α − csc2 α
14.
15. (csc u – 1)( csc u + 1)
16.
17.
1
sec 2 x
+
1
2
csc x
18.
tan α
cot α
csc 2 u
1 + tan 2 u
sin 2 θ − 1
sec θ
sec 2 t − 1
sec 2 t
1 + cot 2 y
1 + tan 2 y
(sec γ − 1)(sec γ + 1)
tan γ
19. sin 4 t + 2 cos 2 t sin 2 t + cos 4 t
20. sin 4 u + 2 cos 2 u − cos 4 u
21. tan 4 α − 2 tan 2 α sec 2 α + sec4 α
22. (1 + tan 2 θ )(1 − sin 2 θ )
23. cos x sin 3 x + sin xcos3 x
24. (1 ‐ cos 2 β )(1 + cot 2 β )
25.
1
cos t
−
sin t cos t sin t
26.
cos γ
cos γ
+
1 ‐ sin γ
1 + sin γ
27.
sin t
1 + cos t
+
1 + cos t
sin t
28.
sin α + sin β
cos α − cos β
+
cos α + cos β
sin α − sin β
In problems 29 to 38, rewrite each trigonometric expression in terms of sines and cosines, and
then simplify the result.
29.
tan x
sec x
30. (cos θ + tan θ sin θ ) cot θ
31.
csc ( -t )
sec ( -t ) cot ( -t )
32.
csc 2 x + sec 2 x
csc 2 x sec 2 x
33.
sec α
csc α ( tan α + cot α )
34.
sin γ + tan γ
1 + sec γ
35.
1 + tan θ
sec θ
36.
cot (−t ) − 1
1 - tan (−t )
37.
tan u + sin u
cot u + csc u
38.
csc β
csc β
+
csc β + tan β
csc β − tan β
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In problems 39 to 44, rewrite each expression in terms of the indicated function only.
2
39. sec θ tan θ in terms of cos θ
41.
43.
1 + cot 2 x
cot 2 x
in terms of cos x
sin(− α ) + tan(− α )
1 + sec(− α )
40.
sin t + cot t cos t
cot t
42.
csc 2γ + sec 2γ
cscγ sec γ
in terms of sec t
in terms of tan γ
in terms of sin α
44. ( cot u + csc u ) ( tan u – sin u ) in terms of sec u
In problems 45 to 96, show that each trigonometric equation is an identity.
45. sin θ sec θ = tan θ
46. cos α tan α csc α = 1
47. tan x cos x = sin x
48. sin β cot β sec = 1
49. csc(– t ) tan (– t ) = sec t
50. sin(– u ) = sin u csc(– u )
51. tan α sin α + cos α = sec α
52.
53.
sin β
cos β
+
=1
csc β
sec β
2
sec x csc x
tan x + cot x
=1
54. 2 – sin 2θ = 1 + cos 2θ
55. cos 2t ( 1 + tan 2 t ) = 1
56. sec 2υ ( 1 – sin 2υ ) = 1
57. sec 2 w cot 2 w – cos 2 w csc 2 w = 1
58. tan 4 u – sec 4 u = 1 – 2 sec 2 u
59. sin 2θ cot 2θ + cos 2θ tan 2θ = 1
60. cot 2γ – cos 2γ = cot 2γ cos 2γ
61. sin 2v + tan 2v + cos 2v = sec 2v
62. 2 csc β – cot β cos β = sin β + csc β
63. sin 2 x + cos 2 x ( 1 – tan 2 x ) = cos 2x
64. sin 4 t – cos 4t + 2 sin 2t cot 2t = 1
65.
67.
tan θ
1 + tan θ
2
=
sin θ
sec θ
sin 2 t
+ cos t = sec t
cos t
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66.
cos 2 s
1
+
= csc s
sin s
csc s
68.
1
= sin θ cos θ
tan θ + cot θ
69.
sin 3 t
+ sin t cos t = tan t
cos t
71.
sin β + cos β
sec β + csc β
=
sin β − cos β
sec β − csc β
73.
sin x cos x
1 − 2sin 2 x
=
70.
⎝ csc t
74.
1 − tan 2 x
tan u sin u
tan u − sin u
=
77.
1 − cot(−α )
=
1 − tan(−α )
cot α
sin u
1 − cos u
79. (cot β + csc β )2 =
=
cot x − 1
cot x + 1
2
1 + sin t
1 + csc t ⎞
2
72. ⎛⎜
⎟ sec t =
tan x
75.
csc x − sec x
csc x + sec x
1 − sin t
⎠
(1 − cot y ) 2 + 2 sin y cos y = 1
2
csc y
76. (secγ − tanγ )2 =
sec β + 1
sec β − 1
1 − sinγ
1 + sinγ
78.
1
csc x − cot x
=
2
1
−
sin x csc x + cot x
80.
csc 2t + sec 2t
csc t sec t
= cot t + tan t
81.
cos x
1 − cos x tan x
=
1 + cos x tan x
cos x
82.
cot β − csc β + 1
cot β + csc β − 1
83.
sin θ
sin θ
−
=2
cotθ + csc θ cotθ − csc θ
84.
tan x − tan y
1 + tan x tan y
85.
1
1
1
+
=
2
2
2
sin t cos t sin t − sin 4 t
86. cos 6θ − sin 6θ = (2cos 2θ − 1)(1 − sin 2θ cos 2θ )
87.
cos(−α )
sin( −α )
−
= sinα + cosα
1 + tan(−α ) 1 + cot(−α )
88. ( 1 + tan β + sec β ) ( 1 + cot β – csc β ) = 2
89.
sec u
1 + cos u
+
= 2 csc u
csc u (1 + sec u )
sin u
90.
91. ( 1 + tan β + cot β ) ( cos β – sin β ) =
csc β
secβ
−
2
sec β csc 2 β
93. ( 1 – cot w ) 2 ( 1 + cot w ) 2 + 4 cot 2 w = csc4 w
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2
2
sec x tan x
92.
94.
95. ( 1 + sin ω t + cos ω t ) 2 = 2 ( 1 + sin ω t ) ( 1 + cos ω t )
sec 4 x + tan 4 x
=
=
=
sin β
1 + cos β
cot y − cot x
1 + cot x cot y
cos 4 x
sin 2 x
+2
cos γ
sec γ − tan γ
=
1 + sin γ
sec γ + tan γ
cot α + cscα
+ secα = 0
sin α + cot(−α ) + csc(−α )
cot x ⎞ ⎛ tan x ⎞
⎛ sec x ⎞
96. ⎛⎜
⎟+⎜
⎟ = 1+ ⎜
⎟
⎝ 1 − tan x ⎠ ⎝ 1 − cot x ⎠
⎝ sin x ⎠
In problems 97 to 103, show that the given trigonometric equation is not a trigonometric identity.
97. sin θ – sec θ = tan θ – 1
99.
101.
sin t + tan t
= tan t
cos t + tan t
1 + sin 2u = 1 + sin u
98. ( sin x + cos x ) 2 = 1
100. cos ( γ + π ) = cos γ
102. ln ( sin x ) = sin( ln x )
103. sin( t 2 ) = sin 2 t
104. Give an example of a trigonometric equation that is true for three different values of
the variable but isn’t an identity.
30