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Pre-Calc Exam Notes 69

Pre-Calc Exam Notes 69 - Basic Trigonometric Identities...

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Basic Trigonometric Identities Section 3.1 69 Example3.5 Prove that tan 2 θ + 2 1 + tan 2 θ = 1 + cos 2 θ . Solution: Expand the left side: tan 2 θ + 2 1 + tan 2 θ = ( tan 2 θ + 1 ) + 1 1 + tan 2 θ = sec 2 θ + 1 sec 2 θ (by (3.10)) = sec 2 θ sec 2 θ + 1 sec 2 θ = 1 + cos 2 θ When trying to prove an identity where at least one side is a ratio of expressions, cross- multiplying can be an effective technique: a b = c d if and only if ad = bc Example3.6 Prove that 1 + sin θ cos θ = cos θ 1 sin θ . Solution: Cross-multiply and reduce both sides until it is clear that they are equal: (1 + sin θ )(1 sin θ ) = cos θ · cos θ 1 sin 2 θ = cos 2 θ By (3.5) both sides of the last equation are indeed equal. Thus, the original identity holds. Example3.7 Suppose that a cos θ = b and c sin θ = d for some angle θ and some constants a , b , c , and d . Show that a 2 c 2 = b 2 c 2 + a 2 d 2 . Solution:
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