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770 Foundations of Trigonometry
10.4 Trigonometric Identities
In Section 10.3, we saw the utility of the Pythagorean Identities in Theorem 10.8 along with the
Quotient and Reciprocal Identities in Theorem 10.6. Not only did these identities help us compute
the values of the circular functions for angles, they were also useful in simplifying expressions
involving the circular functions. In this section, we introduce several collections of identities which
have uses in this course and beyond. Our first set of identities is the ‘Even / Odd’ identities.1
Theorem 10.12. Even / Odd Identities: For all applicable angles θ,
• cos(−θ) = cos(θ)
• sec(−θ) = sec(θ)
• sin(−θ) = − sin(θ)
• csc(−θ) = − csc(θ)
• tan(−θ) = − tan(θ)
• cot(−θ) = − cot(θ)
In light of the Quotient and Reciprocal Identities, Theorem 10.6, it suffices to show cos(−θ) = cos(θ)
and sin(−θ) = − sin(θ). The remaining four circular functions can be expressed in terms of cos(θ)
and sin(θ) so the proofs of their Even / Odd Identities are left as exercises. Consider an angle θ
plotted in standard position. Let θ0 be the angle coterminal with θ with 0 ≤ θ0 < 2π. (We can
construct the angle θ0 by rotating counter-clockwise from the positive x-axis to the terminal side
of θ as pictured below.) Since θ and θ0 are coterminal, cos(θ) = cos(θ0) and sin(θ) = sin(θ0).
x
y
1
1
θ
θ0
x
y
1
1
θ0
−θ0
P(cos(θ0), sin(θ0))
Q(cos(−θ0), sin(−θ0))
We now consider the angles −θ and −θ0. Since θ is coterminal with θ0, there is some integer k so
that θ = θ0 + 2π · k. Therefore, −θ = −θ0 − 2π · k = −θ0 + 2π · (−k). Since k is an integer, so is
(−k), which means −θ is coterminal with −θ0. Hence, cos(−θ) = cos(−θ0) and sin(−θ) = sin(−θ0).
Let P and Q denote the points on the terminal sides of θ0 and −θ0, respectively, which lie on the
Unit Circle. By definition, the coordinates of P are (cos(θ0),sin(θ0)) and the coordinates of Q are
(cos(−θ0),sin(−θ0)). Since θ0 and −θ0 sweep out congruent central sectors of the Unit Circle, it
1As mentioned at the end of Section 10.2, properties of the circular functions when thought of as functions of
angles in radian measure hold equally well if we view these functions as functions of real numbers. Not surprisingly,
the Even / Odd properties of the circular functions are so named because they identify cosine and secant as even
functions, while the remaining four circular functions are odd. (See Section 1.6.)