Reciprocal and quotient identities · derivation ·proof
Where tan θ = sin θ / cos θ comes from
Cancel the hypotenuse. If you need to remove tangent from an expression, the question to ask is 'what is tangent made of?' — two ratios whose denominator cancels.
tan θ = sin θ / cos θ
Building it step by step
- tan θ = opp/adjDefinition.
- opp/adj = (opp/hyp)/(adj/hyp)Multiply top and bottom by 1/hyp.
- = sin θ/cos θRecognise both ratios.
How to recall it under pressure
Re-derive it from the first line rather than searching memory: start from tan(x) and do the one algebraic move the derivation above makes. If you need the statement itself, it is on the quotient identity (tangent) page.