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§6-2 Sum, Difference, and Cofunction Identities
In this section, we will learn following identities:
Summary of Identities Sum Identities cos(x + y) = cos x cos y − sin x sin y sin(x + y) = sin x cos y + cos x sin y
tan(x + y) =
tan x + tan y 1 − tan x tan y Difference Identities cos(x − y) = cos x cos y + sin x sin y sin(x − y) = sin x cos y − cos x sin y
tan(x − y) =
tan x − tan y 1 + tan x tan y Cofunction Identities (Replace π/2 with 90o^ if x is in degrees.)
cos(
π 2
− x) = sin x sin(
π 2
− x) = cos x tan(
π 2
− x) = cot x
We will prove difference identity for cosine:
cos(x − y) = cos x cos y + sin x sin y (1)
And other identities can be readily verified from this particular one. At first, we assume that x and y are in the interval (0, 2 π) and x > y > 0. We associate x and y with arcs and angles on the unit circle as indicated in Figure 1. We form a triangle AOB, with A = (cos y, sin y), B = (cos x, sin y). Now, we rotate the triangle AOB clockwise about the origin until the terminal point A coincides with D(1, 0), then terminal B will be at C = (cos(x − y), sin(x − y)). Since the rotation preserves lengths, we have
d(A, B) = d(C, D) √ (c − a)^2 + (d − b)^2 =
(1 − e)^2 + (0 − f )^2 (c − a)^2 + (d − b)^2 = (1 − e)^2 + f 2 c^2 − 2 a c + a^2 + d^2 − 2 d b + b^2 = 1 − 2 e + e^2 + f 2 (c^2 + d^2 ) + (a^2 + b^2 ) − 2 a c − 2 d b = 1 − 2 e + (e^2 + f 2 )
Since points A, B and C are on unit circles, we have c^2 + d^2 = 1, a^2 + b^2 = 1 and e^2 + f 2 = 1, and we have
e = a c + b d 1
6 º
R
y
x
O D(1,^ 0)
B(cos x, sin x)
A(cos y, sin y)
a (^) b
c (^) d
x − y
R
6 ™ x^ −^ y
D(1, 0)
C(cos(x − y), sin(x − y))
e f
Figure 1
Replacing e, a, c, b, d with cos(x−y), cos y, cos x, sin y and sin x, respectively(Figure 1), we obtain
cos(x − y) = cos y cos x + sin y sin x = cos x cos y + sin x sin y
Thus we prove the difference identity for cosine for x and y are in the interval (0, 2 π) and x > y > 0. It then follows easily, by periodicity and basic identities, that the identity holds for all real numbers and angles in radian or degree measure.
Verify other identities
Sum identity for cosine: Replace y with −y in equation (1), we obtain the sum identity for cosine:
cos(x + y) = cos x cos y − sin x sin y (2)
Cofunction identities: Set x = π 2 in the identity (1), we have
cos
( (^) π
2
− y
= cos
π 2
cos y + sin
π 2
sin y
= (0) (cos y) + (1) (sin y) = sin y
Hence, we have the cofunction identity for cosine:
cos
( (^) π 2
− y
= sin y (3)
Cofunction identities for sine: Now, let y = π/ 2 − x in equation (3), we have
cos
[ (^) π
2
( (^) π
2
− x
)]
= sin
( (^) π
2
− x
cos x = sin
( (^) π
2
− x
Replace y in equation (8) with −y, we obtain the sum identity for tangent:
tan(x + y) =
tan x + tan y 1 − tan x tan y
Example Simplify cos(x − π) using the difference identity
Solution.
cos(x − π) = cos x cos π + sin x sin π = (cos x)(−1) + (sin x)(0) = − cos x
Hence, we have cos(x − π) = − cos x § Example Find the exact value of tan 75o.
Solution. Since 75o^ = 45o^ + 30o, we can use the sum identity for tangent with x = 45o^ and y = 30o^ to find the exact value of tan 75o.
tan(x + y) =
tan x + tan y 1 − tan x tan y
tan(45o^ + 30o) =
tan 45o^ + tan 30o 1 − tan 45o^ tan 30o
=
Example Verify the identity:
tan x + cot y =
cos(x − y) cos x sin y
Solution. cos(x − y) cos x sin y
cos x cos y + sin x sin y cos x sin y
=
cos x cos y cos x sin y
sin x sin y cos x sin y = cot y + tan y = tan x + cot y § Further reading: Section 6-2. Exercise: Ex 6-2: 15 - 20, 21-28, 33-46, 29, 31 (refer to the example 4).
