7.5 The Sum and Difference Identities
The previous sections dealt with trig functions in one variable. This section will address
trig functions in two variables. We will only consider sine, cosine, and tangent for
brevity. Watch Mr. Burger to see how these identities can easily be remembered.
sin(α + β) = sin α cos β + cos α sin β
sin(α – β) = sin α cos β – cos α sin β
cos(α + β) = cos α cos β – sin α sin β
cos(α – β) = cos α cos β + sin α sin β
tan   tan 
1  tan  tan 
tan   tan 
tan(α – β) =
1  tan  tan 
tan(α + β) =
Example Simplify sin (x + 3π/2) using a sum identity.
Sum identity for sine is sin(α + β) = sin α cos β + cos α sin β
sin (x + 3π/2) = sin x cos 3π/2 + cos x sin 3π/2
= sin x (0) + cos x (-1)
= 0 – cos x
= -cos x
Example Find the value of tan 75˚ in exact radical form.
We want to use one of the tangent identities. To figure which one to use we want to see
what basic angles in which you know the tangent of has a sum of 75˚.
75 = 45 + 30
tan   tan 
Therefore, we want to use the sum identity for tangent: tan(α + β) =
1  tan  tan 
tan 45  tan 30
tan 75˚ = tan(45˚ + 30˚) =
1  tan 45 tan 30
3
3 3
1
3 3 3 3
3
3

=
=
=
3
3
 3   3 3 
1  1
 

 3   3 
3 3
3
93 3
=

3
3 3
93 3
9  3 3 9  3 3 81  54 3  27 108  54 3
=
=
=
= 2 3

81  27
54
93 3 93 3
=
Example Simplify the identity: cos (π/9) cos (7π/18) – sin(π/9) sin (7π/18)
This problem correlates with sum identity for cosine:
cos(α + β) = cos α cos β – sin α sin β
cos (π/9) cos (7π/18) – sin(π/9) sin (7π/18) = cos(π/9 + 7π/18) =
= cos (9π/18) = cos(π/2) = 0
Try the following:
1. Simplify cos (π/3 – x) using a difference identity.
2. Find the value of sin 135˚ in exact radical form.
3. Simplify the identity:
tan(19 / 36)  tan(5 /18)
1  tan(19 / 36) tan(5 /18)
Answers: ½ cos x +  3  sin x ;
 2
2
; 1
2