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The reciprocal trigonometric functions are secant (1/cos), cosecant (1/sin) and cotangent (1/tan). This topic covers their graphs, the identities 1 + tan²x = sec²x and 1 + cot²x = cosec²x, and using them in proofs and equations. It appears across A-Level and iAL Pure Maths with the major exam boards. Work through it free on eClassroom, with clear worked examples and instantly-marked questions that build your confidence for the exam.
How you do it
Sec is one over cos, cosec is one over sin and cot is one over tan. The identities follow by dividing the Pythagorean identity by cos squared or sin squared. Rewriting in terms of sin and cos is usually the quickest route through a problem.
Before you start
Bring trig ratios and identities with you. From here the path runs to harder identities and calculus.
What you learn
From them come two powerful identities, sec ²θ≡ 1+ tan ²θ and operatornamecosec²θ≡ 1+ cot ²θ, which turn awkward equations into quadratics. Each is the reciprocal of a familiar function. Both come from sin ²θ+ cos ²θ≡ 1 — you can derive them, so there is nothing extra to memorise. An equation mixing sec ² with tan (or operatornamecosec² with cot) becomes a quadratic once you substitute the identity. The course works through sec, cosec and cot, sketching sec, cosec and cot, the two new identities, equations that become quadratics and prove, then solve. Every section has instant-feedback practice questions matched to Edexcel International A-Level and 9MA0 Pure, and the course finishes with full exam-style questions you mark yourself.
Related Topics
Reciprocal Trig Functions is a Trigonometry topic in the A-level syllabus, covered here at the AS year. The course is built from 5 concept sections, 5 worked examples and 5 check points before the exam-style set. Nearby topics in the same strand are Sine & Cosine Rules, Solving Trig Equations, Trig Graphs & Transformations and Trig Identities (C12). For unlimited practice, the Trigonometry revision engine builds new questions on demand and records your accuracy.
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