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 = cost
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 , sinθ )
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 .
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If we now combine the identities above with the quotient identity, tan θ = sin θ cos θ , we
find that
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
25
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.
Example 4.2 ---------------------------- -----------------------------------------------------------Simplify each trigonometric expression by using the fundamental identities.
(a) (csc θ )(cos θ )
1
(csc θ )(cos θ ) =
sin θ
cos θ =
cos θ
sin θ
= cot θ
(b) tan 2 t − sec2 t
2
2
2
2
Because 1 + tan θ = sec θ , it follows that tan θ − sec θ = − 1 .
(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
=
1
sin t
–
cos t
sin t
1
=
1
–
sin t
cos t
sin t
cos t
cos t
=
(b)
1
sin t
2
– cos t =
sin t
2
1 - cos t
sin t
2
= sin t = sin t
sin t
csc 2 x sec 2 x
csc 2 x + sec 2 x
26
⎛ 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 ⎞⎛ 1 ⎞
2 ⎟⎜
2 ⎟
⎝ sin x ⎠⎝ cos x ⎠
1
1 ⎞
2
2 ⎛
sin x cos x⎜
+
⎟
2
⎝ sin x cos2 x ⎠
2
2
sin x cos x⎜
= 1.
_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
The Pythagorean identity cos 2 θ + sin 2 θ = 1 can be written as
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.
2
cot θ csc θ =
cos θ
sin θ
.
1
2
sin θ
=
cos θ
3
sin θ
2
=
± 1 − sin θ
3
sin θ
.
_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________
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 –
2
2
a −u
π
2
< θ <
π
2
, so that cos θ > 0.
= a 2 − ( a sin θ ) 2 = a 2 − a 2 sin 2 θ
= a 2 (1 − sin 2 θ ) = a 2 cos 2 θ
= a cos θ .
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Section 4 Problems---------------------- -----------------------------------------------------------In problems 1 to 6, use the even-odd identities to simplify each expression.
1. sin (– θ ) cos (– θ )
2. cot 2 (−u ) + 1
4. cos( –x ) sec x
3. tan ( t ) + tan ( – t )
1 + csc ( −α )
5. 1 − cot (− β )
6. [ 1 + sin γ ][ 1 + sin(– γ )]
In problems 7 to 28, use the fundamental identities to simplify each expression.
7. sec θ sin θ
8. 1 +
9. cot υ sec υ
10.
csc β
sec β
12.
11.
13. cot 2 α − csc 2 α
14.
15. (csc u – 1)( csc u + 1)
16.
17.
1
2
sec x
+
1
18.
2
csc x
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 + sec 4 α
22. (1 + tan 2 θ )(1 − sin 2 θ )
3
3
2
25.
1
cos t
−
sin t cos t sin t
27.
sin t
1 + cos t
+
1 + cos t
sin t
2
24. (1 ‐ cos β )(1 + cot β )
23. cos x sin x + sin xcos x
26.
28.
cos γ
cos γ
+
1 ‐ sin γ
1 + sin γ
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
28
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
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.
2
sin β
cos β
=1
+ csc β sec β 2
2
2
2
2
54. 2 – sin θ = 1 + cos θ 2
2
2
56. sec υ ( 1 – sin υ ) = 1 55. cos t ( 1 + tan t ) = 1 2
sec x csc x
=1
tan x + cot x
2
57. sec w cot w – cos w csc w = 1
4
4
2
58. tan u – sec u = 1 – 2 sec u 29
2
2
2
2
2
59. sin θ cot θ + cos θ tan θ = 1
2
2
2
2
61. sin v + tan v + cos v = sec v
2
2
2
2
63. sin x + cos x ( 1 – tan x ) = cos x 65.
tan θ
1 + tan 2θ
=
2
2
sin θ
sec θ
62. 2 csc β – cot β cos β = sin β + csc β
4
4
2
66.
1
cos 2 s
+
= csc s
csc s
sin s
sin 2 t
+ cos t = sec t
cos t
68.
1
= sin θ cos θ
tan θ + cot θ
69.
sin 3 t
+ sin t cos t = tan t
cos t
70.
csc x − sec x
csc x + sec x
71.
sin β + cos β
sec β + csc β
=
sin β − cos β
sec β − csc β
sin x cos x
1 − 2sin 2 x
=
tan x
1 − tan 2 x
75.
tan u sin u
tan u − sin u
=
sin u
1 − cos u
77.
1 − cot(−α )
=
1 − tan(−α )
cot α
79. (cot β + csc β ) =
2
sec β + 1
sec β − 1
2
64. sin t – cos t + 2 sin t cot t = 1
67.
73.
2
60. cot γ – cos γ = cot γ cos γ
=
cot x − 1
cot x + 1
2
1 + sin t
1 + csc t ⎞
2
72. ⎛⎜
⎟ sec t =
⎝ csc t
74.
1 − sin t
⎠
(1 − cot y ) 2 + 2 sin y cos y = 1
2
csc y
76. (secγ − tanγ )2 =
78.
1
csc x − cot x
80.
csc 2 t + sec 2 t
csc t sec t
=
1 − sinγ
1 + sinγ
2
1
−
sin x csc x + cot x
= 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
87.
cos(−α )
sin( −α )
−
= sinα + cosα
1 + tan(−α ) 1 + cot(−α )
86. cos 6θ − sin 6θ = (2cos 2θ − 1)(1 − sin 2θ cos 2θ )
30
=
=
sin β
1 + cos β
cot y − cot x
1 + cot x cot y
88. ( 1 + tan β + sec β ) ( 1 + cot β – csc β ) = 2
89.
90.
sec u
1 + cos u
+
= 2 csc u
csc u (1 + sec u )
sin u
sec 4 x + tan 4 x
sec 2 x tan 2 x
=
cos 4 x
sin 2 x
+2
91. ( 1 + tan β + cot β ) ( cos β – sin β ) =
92.
csc β
sec 2 β
−
secβ
csc 2 β
cos γ
sec γ − tan γ
=
sec γ + tan γ 1 + sin γ
93. ( 1 – cot w ) 2 ( 1 + cot w ) 2 + 4 cot 2 w = csc 4 w
94.
cot α + cscα
+ secα = 0
sin α + cot(−α ) + csc(−α )
95. ( 1 + sin ω t + cos ω t ) 2 = 2 ( 1 + sin ω t ) ( 1 + cos ω t )
96.
cot x
tan x
sec x
+
=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
98. ( sin x + cos x ) 2 = 1
sin t + tan t
= tan t
cos t + tan t
100. cos ( γ + π ) = cos γ
99.
101. 1 + sin 2 u = 1 + sin u
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.
31