Identities Summary

Trigonometry · Axiom Academy

SUMMARY Trigonometric Identities Let's review the fundamental equations that are true for all valid angles, providing powerful tools for simplification and elegant problem-solving. Reciprocal Identities: csc θ = 1/sin θ , sec θ = 1/cos θ , cot θ = 1/tan θ Quotient Identities: tan θ = sin θ/cos θ and cot θ = cos θ/sin θ Key Insight: These identities allow conversion between different trigonometric functions, essential for simplification Common Strategy: Converting everything to sines and cosines often reveals hidden patterns and simplifies proofs Sine: sin(A ± B) = sin A cos B ± cos A sin B Cosine: cos(A ± B) = cos A cos B ∓ sin A sin B Tangent: tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B) Critical Note: sin(A + B) ≠ sin A + sin B — angle addition requires these special formulas! Application: Calculate exact values for non-special angles like 75° = 45° + 30° or 15° = 45° - 30° Proving an Identity: (1 - cos x)(1 + cos x) = sin²x Identify the strategy: The left side is a product that looks like a difference of squares: (a - b)(a + b) = a² - b² Apply algebraic technique: (1 - cos x)(1 + cos x) = 1² - cos²x = 1 - cos²x Recognize the identity: From the Pythagorean identity sin²x + cos²x = 1, we can rearrange to get sin²x = 1 - cos²x Make the substitution: 1 - cos²x = sin²x Conclusion: Left side equals right side, so the identity is proven ✓ Key insight: Combining algebraic techniques with known identities is the essence of proving trigonometric identities

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