Pythagorean & reciprocal identities
sin²θ + cos²θ = 1 comes straight from Pythagoras on the unit circle, and dividing it by cos²θ or sin²θ gives 1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ. Together with the reciprocal definitions sec = 1/cos, csc = 1/sin and cot = 1/tan, these three identities handle most simplification questions.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Convert everything to sine and cosineWhen stuck, rewriting sec, csc, cot and tan in terms of sin and cos almost always reveals the route.
- Spot the Pythagorean patterns1 − sin²θ is cos²θ; sec²θ − 1 is tan²θ. Recognising these is the main skill.
- Combine fractions over a common denominatorThen simplify the numerator using an identity.
- Track the sign when taking square rootscos θ = ±√(1 − sin²θ), and the quadrant decides which sign applies.
Worked example
Given sin θ = 3/5 with θ in the second quadrant, find cos θ and tan θ exactly.
- cos²θ = 1 − sin²θ = 1 − 9/25 = 16/25.
- So cos θ = ±4/5.
- In the second quadrant cosine is negative, so cos θ = −4/5.
- tan θ = sin θ/cos θ = (3/5)/(−4/5).
Answer. cos θ = −4/5 and tan θ = −3/4.
Where marks get dropped
These are the specific errors that cost credit on pythagorean & reciprocal identities questions — QED's rubric penalises each of them separately.
- Taking the positive square root automatically. The quadrant determines the sign, and ignoring it is the standard lost mark here.
- Writing sin²θ as sin(θ²). The notation means (sin θ)², a completely different quantity.
- Misremembering the variants. Dividing by cos² gives 1 + tan² = sec²; dividing by sin² gives 1 + cot² = csc².
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Pythagorean & reciprocal identities — frequently asked questions
How do I remember the variants?
Derive them rather than memorising: divide sin² + cos² = 1 by cos²θ for the tan/sec version, and by sin²θ for the cot/csc version. Two lines each.
What is sec θ?
1/cos θ. It is undefined wherever cos θ = 0, which gives the same asymptotes as tangent.
Why does the quadrant matter so much?
Because sin²θ + cos²θ = 1 determines only the magnitude of the other ratio. The sign comes entirely from the quadrant.
The rest of Trigonometry
Triangles, the unit circle, identities and periodic functions. Each subtopic below has its own method, worked example and mark-losing traps.
- 1Right-triangle ratios: sin, cos & tan
- 2The unit circle & exact values
- 3Radians, degrees & arc length
- 4Graphs of sin, cos & tan
- 5Pythagorean & reciprocal identities
- 6Angle-sum, difference & double-angle formulas
- 7Solving trigonometric equations
- 8The sine rule & the cosine rule
- 9Triangle area, sectors & segments
- 10Inverse trigonometric functions
- 11Proving trigonometric identities
- 12Polar coordinates & converting to Cartesian
- 13Modelling periodic phenomena with sinusoids
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