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Unformatted text preview: gles 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 1.7.) 656 Foundations of Trigonometry sin(−θ0 ) = − sin(θ0 ). Since the cosines and sines of θ0 and −θ0 are the same as those for θ and
−θ, respectively, we get cos(−θ) = cos(θ) and sin(−θ) = − sin(θ), as required. The Even / Odd
Identities are readily demonstrated using any of the ‘common angles’ noted in Section 10.2. Their
true utility, however, lies not in computation, but in simplifying expressions involving the circular
functions. Our next batch of identities makes heavy use of the Even / Odd Identities.
Theorem 10.13. Sum and Diﬀerence Identities for Cosine: For all angles α and β ,
• cos(α + β ) = cos(α) cos(β ) − sin(α) sin(β )
• cos(α − β ) = cos(α) cos(β ) + sin(α) sin(β )
We ﬁrst prove the result for diﬀerences. As in the proof of the Even / Odd Identities, we can reduce
the proof for general angles α and β to angles α0 and β0 , coterminal to α and β , respectively, each
of which measure between 0 and 2π radians. Since α and α0 are cot...
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