Fundamental identities
Objective. Use tan θ=sin θ/cos θ and sin²θ+cos²θ=1.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
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Learn: the key idea
Fundamental trigonometric identities connect the ratios algebraically: tan θ=sin θ/cos θ and sin²θ+cos²θ=1 hold for every angle θ where defined, letting you simplify expressions or prove further identities. For example, to simplify sin θ/cos θ × cos θ, the cos θ terms cancel to leave sin θ directly. Using sin²θ+cos²θ=1, you can rewrite sin²θ as 1−cos²θ to simplify expressions mixing both functions. Always substitute identities to reduce the number of different trig functions present before attempting further simplification.
Worked example
Simplify the expression cos²θ + sin²θ − tan θ sin θ cos θ using trigonometric identities.
- 1Apply sin²θ+cos²θ=1 to the first two terms
- 2Rewrite tan θ as sin θ/cos θ in the last term
- 3Simplify (sin θ/cos θ)×sin θ×cos θ to sin²θ
- 4Combine to get 1 − sin²θ, which equals cos²θ
Answer: cos²θ
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Practice
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Start practisingCommon mistake. Students assume sin²θ+cos²θ=1 means sin θ+cos θ=1, incorrectly removing the squares; the identity applies strictly to the squared terms and cannot be simplified by taking square roots term by term.
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