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**Unformatted text preview: **ce of identities in trigonometry. Our next task is to use use the
Reciprocal and Quotient Identities found in Theorem 10.6 coupled with the Pythagorean Identity
found in Theorem 10.1 to derive new Pythagorean-like identities for the remaining four circular
functions. Assuming cos(θ) = 0, we may start with cos2 (θ) + sin2 (θ) = 1 and divide both sides
sin2
by cos2 (θ) to obtain 1 + cos2(θ) = cos1(θ) . Using properties of exponents along with the Reciprocal
2
(θ)
and Quotient Identities, reduces this to 1 + tan2 (θ) = sec2 (θ). If sin(θ) = 0, we can divide both
sides of the identity cos2 (θ) + sin2 (θ) = 1 by sin2 (θ), apply Theorem 10.6 once again, and obtain
cot2 (θ) + 1 = csc2 (θ). These three Pythagorean Identities are worth memorizing, and they are
summarized in the following theorem. 640 Foundations of Trigonometry Theorem 10.8. The Pythagorean Identities:
• cos2 (θ) + sin2 (θ) = 1.
• 1 + tan2 (θ) = sec2 (θ), provided cos(θ) = 0.
• cot2 (θ) + 1 = csc2 (θ), provided sin(θ) = 0.
Trigonometric identities play an important role in not just Trigonometry, but in Calculus as well.
We’ll use them in this book to...

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