68 Chapter 3 • Identities §3.1 Example 3.3 Prove that tan θ + cot θ = sec θ csc θ . Solution: We will expand the left side and show that it equals the right side: tan θ + cot θ = sin θ cos θ + cos θ sin θ (by (3.1) and (3.2)) = sin θ cos θ · sin θ sin θ + cos θ sin θ · cos θ cos θ (multiply both fractions by 1) = sin 2 θ + cos 2 θ cos θ sin θ (after getting a common denominator) = 1 cos θ sin θ (by (3.3)) = 1 cos θ · 1 sin θ = sec θ csc θ In the above example, how did we know to expand the left side instead of the right side? In general, though this technique does not always work, the more complicated side of the identity is likely to be easier to expand. The reason is that, by its complexity, there will be more things that you can do with that expression. For example, if you were asked to prove
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