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LESSON 7
TRIGONOMETRIC IDENTITIES
TRIGONOMETRIC IDENTITIES
A trigonometric identity is an equation involving
trigonometric functions that hold for all values of the
argument, typically chosen to be θ. In other words, an
identity is an equation that is true for all of its domain
values.
FUNDAMENTAL TRIGONOMETRIC IDENTITIES
Table 1: Fundamental Trigonometric Identities
VERIFICATION OF TRIGONOMETRIC IDENTITIES
To verify an identity, we show that one side of the
identity can be rewritten in an equivalent form that is
identical to the other side. There is no one method that
can be used to verify every identity; however the
following guidelines should prove useful.
Guidelines for Verifying Trigonometric Identities
• If one side of the identity is more complex than the other,
then it is generally best to try first to simplify the more
complex side until it becomes identical to the other side.
• Perform indicated operations such as adding fractions or
squaring a binomial. Also be aware of any factorization
that may help you to achieve your goal of producing the
expression on the other side.
• Use previously established identities that enable you to
rewrite one side of the identity in an equivalent form.
• Rewrite one side of the identity so that it involves only sine
and/or cosines.
• Rewrite one side of the identity in terms of a single
trigonometric function.
• Multiplying both the numerator and the denominator of a
fraction by the same factor (such as the conjugate of the
denominator or the conjugate of the numerator) may get
you closer to your goal.
• Keep your goal in mind. Does it involve products, quotients,
sums, radicals, or powers? Knowing exactly what your goal
is may provide the insight you need to verify the identity.
EXAMPLES
Verify the following identities.
a) ta n s ec s i n  ta n2
h) s i n4   cos4   s i n2   cos2 
s i n
cot x  ta nx
b)
 cs c   cot 
i ) s ecx 
1  cos 
cs c x
1  ta n3
1  s i nx 1  s i nx
2
c)
 1  ta n  ta n  j)

 4 s ecx ta nx
1  ta n
1  s i nx 1  s i nx
d) s i n  cos s i n  cos   1  2 cos2 
cos x ta nx  2 cos x  ta nx  2
e)
 cos x  1
ta nx  2
s i n2x cos x  cos3x  s i n3x cos x  s i nx cos3x
f)
1  s i n2 x
1  s i nx  cos x
cos x
g)

1  s i nx  cos x 1  s i nx
SUM AND DIFFERENCE IDENTITIES
cos (α – β) = cos α cos β + sin α sin β
cos (α + β) = cos α cos β – sin α sin β
sin (α – β) = sin α cos β – cos α sin β
sin (α + β) = sin α cos β + cos α sin β
EXAMPLES
1. Find the exact value of each trigonometric expression.
a) cos 750
b) sin (π/12)
c) tan 1950
2. Write each expression as a single trigonometric expression.
a) sin (–x) cos 3x – cos (–x) sin 3x
b) cos 4x cos (–2x) – sin 4x sin (–2x)
3. Find the exact value of a) cos (β – α) and b) sin (α + β), if
sin α = 24/25, α in QII, and cos β = – (4/5), β in QIII.
4. Verify the following identities:
a) cos (θ + π) = – cos θ
b) sin (α – β) – sin (α + β) = – 2 cos α sin β
DOUBLE – ANGLE IDENTITIES
Sine
Cosine
sin 2α = 2 sin α cos α cos 2α = cos2 α – sin2 α
cos 2α = 1 – 2 sin2 α
cos 2α = 2 cos2 α – 1
POWER–REDUCING IDENTITIES
Tangent
EXAMPLES
1. Write each trigonometric expression in terms of a single
trigonometric function .
a) 2 sin 3θ cos 3θ
b) cos2(x + 2) – sin2(x + 2)
2. If cos α = 24/25; 2700 < α < 3600, find a) sin 2α b) cos 2α
and c) tan 2α
3. Use the power–reducing identities to write each
trigonometric expression in terms of the first power of one
or more cosine funtions.
a) cos4 α
b) sin2 x cos4 x
4. Verify :
1
1 2
2
2
 csc x
a) cos 8x = cos 4x – sin 4x
b)
1  cos 2x 2
HALF – ANGLE IDENTITIES
Sine
Cosine
Tangent

1  cos 
tan  
2
2
tan

sin 

2 1  cos 
tan
 1  cos 

2
sin 
EXAMPLES
1. Use half – identities to find the exact value of each
trigonometric expression.
a) cos 1050
b) sin (3π/8)
c) cot 67.50
2. Find the exact value of the sine, cosine, and tangent of (α/2)
given the following information.
a) cos α = 12/13, 00 < α < 900
b) csc α = – (5/3), α is in Quadrant IV
3. Verify the following identities :
2 x
2 x
a) cos  sin  cos x
2
2
sinx
2 x
b) 2 csc x cos 
2 1  cos x
PRODUCT – TO – SUM IDENTITIES
1
sin cos   sin    sin  
2
1
cos  sin  sin    sin  
2
1
cos  cos   cos     cos   
2
1
sin sin  cos     cos   
2
SUM – TO – PRODUCT IDENTITIES
x  y
x  y
s inx  cos y  2 s in
 cos 

 2 
 2 
x  y
x  y
cos x  cos y  2 cos 
 cos 

 2 
 2 
x  y x  y
s inx  s iny  2 cos 
 s in

 2   2 
x  y x  y
cos x  cos y  2 sin
 sin

 2   2 
EXAMPLES
1. Write each expression as a sum or difference of sines
and/or cosines.
a) 2 sin 4x sin 2x
b) 4 cos (– x) cos 2x
c) sin (3x/2) sin (5x/2)
d) sin (–πx/4) cos (–πx/2)
2. Write each expression as a product of sines and/or cosines.
a) cos 5x – cos 3x
b) sin (3x/4) + sin (x/2)
c) sin (0.4x) – sin (0.6x) d) cos (–πx/4) + cos (πx/6)
3. Simplify the following trigonometric expressions.
cos 5x  cos 3x
a)
sin5x  sin3x
b) sinx  y  sin(x  y)