Three new names for 1/sin, 1/cos, 1/tan: You already know sin, cos and tan. The HL course adds their reciprocals: same triangle, just flipped.
The three reciprocals
- sec θ = 1/cos θ (goes with cos)
- csc θ = 1/sin θ (also written cosec θ; goes with sin)
- cot θ = 1/tan θ = cos θ / sin θ (goes with tan)
A handy memory trick: The third letters pair up: sec ↔ cos, csc ↔ sin, cot ↔ tan via cos/sin.
Why these identities are true: Start from the one identity you already trust: sin²θ + cos²θ = 1.
Divide every term by cos²θ and you get tan²θ + 1 = sec²θ.
Divide every term by sin²θ instead and you get 1 + cot²θ = csc²θ.
So all three Pythagorean identities come from the same place — you never have to memorise them as separate facts.
IB-style question — evaluate a reciprocal ratio
An acute angle θ has cos θ = 3/5.
Find the exact values of sec θ and tan θ.
Step by step
- sec θ is just the reciprocal of cos θ.
- Find sin θ from sin²θ + cos²θ = 1 (positive root, since θ is acute).
- Then tan θ = sin θ / cos θ.
Final answer
sec θ = 5/3 and tan θ = 4/3.
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The same point, read two ways: Two kinds of relationship link the trig values of related angles: co-function and symmetry.
Co-function (right-angled triangle)
- The two acute angles add to 90°, so one angle's sine is the other's cosine.
- sin(90° − θ) = cos θ
- cos(90° − θ) = sin θ
- In radians, 90° = π/2.
Symmetry (unit circle)
- Reflecting an angle changes signs but not size.
- sin(−θ) = −sin θ (sine is odd)
- cos(−θ) = cos θ (cosine is even)
The golden rule for simplifying: When an expression mixes sec, csc, cot, tan, sin and cos and you're stuck:
Rewrite EVERYTHING in sin and cos, put it over a common denominator, then use sin²θ + cos²θ = 1.
Nine times out of ten the mess collapses to something tiny. This is exactly the move the M25 paper rewards.
IB-style question — rewrite in sin and cos
Write f(x) = 4cot x + sin x as a single fraction over sin x.
Step by step
- Replace cot x by cos x / sin x.
- Common denominator sin x; write sin x as sin²x / sin x.
- Combine over the one denominator.
Final answer
f(x) = (4cos x + sin²x) / sin x.
IB-style question — prove an identity
Prove that sec²x − tan²x = 1.
Step by step
- Start from the Pythagorean identity 1 + tan²x = sec²x.
- Subtract tan²x from both sides to isolate the difference.
- The left side now matches the right — identity proved.
Final answer
sec²x − tan²x = 1, directly from 1 + tan²x = sec²x.