Sec, cosec and cot

Definitions

• sec x = 1/cos x
• cosec x = 1/sin x
• cot x = 1/tan x = cos x/sin x

Tip: the third letter tells you which function: sec is 1/cos, cosec is 1/sin.

Graphs

• y = sec x has asymptotes where cos x = 0 (x = 90°, 270°, ...) and range y ≤ -1 or y ≥ 1.
• y = cosec x has asymptotes where sin x = 0 (x = 0°, 180°, 360°, ...) and the same range.
• y = cot x has asymptotes where sin x = 0 and takes every real value.

Identities

Divide sin² x + cos² x ≡ 1 by cos² x: 1 + tan² x ≡ sec² x.

Divide by sin² x: 1 + cot² x ≡ cosec² x.

Use them to turn an equation into a single function, often a quadratic.

Worked example

Solve sec² θ = 3tan θ - 1 for 0° ≤ θ < 180°.

1. Replace sec² θ with 1 + tan² θ: tan² θ - 3tan θ + 2 = 0.
2. (tan θ - 1)(tan θ - 2) = 0
3. tan θ = 1 gives 45°; tan θ = 2 gives 63.4°.

Answer: θ = 45° or θ = 63.4°

Check yourself

1. Find the exact value of sec 60°.
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   2. cos 60° = 1/2, so sec 60° = 2.

2. θ is acute and cosec θ = 3. Find cot θ, giving your answer to 2 decimal places.
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   2.83. cot² θ = cosec² θ - 1 = 8, so cot θ = sqrt(8) = 2.83 (positive, as θ is acute).

3. What is the range of y = sec x?
   • y ≤ -1 or y ≥ 1
   • -1 ≤ y ≤ 1
   • y ≥ 1
   • y ∈ ℝ
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   y ≤ -1 or y ≥ 1. cos x lies between -1 and 1, so 1/cos x is at least 1 in size.

4. Where are the asymptotes of y = cosec x for 0° ≤ x ≤ 360°?
   • x = 90° and 270°
   • x = 45°, 135°, 225° and 315°
   • x = 180° only
   • x = 0°, 180° and 360°
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   x = 0°, 180° and 360°. cosec x = 1/sin x is undefined where sin x = 0.

5. Solve 2cosec x = 4 for 0° ≤ x ≤ 360°.
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   x = 30° or x = 150°. cosec x = 2, so sin x = 1/2.

6. θ is acute and tan θ = 3/4. Find sec θ. Give your answer as a fraction in its simplest form.
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   5/4. sec² θ = 1 + 9/16 = 25/16, so sec θ = 5/4 (positive, as θ is acute).

7. Solve 2tan² x + sec² x = 4 for 0° ≤ x ≤ 180°.
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   x = 45° or x = 135°. 2tan² x + 1 + tan² x = 4 gives tan² x = 1, so tan x = ±1.

8. Which of these is an identity?
   • cosec² θ ≡ 1 + cot² θ
   • sec² θ ≡ 1 - tan² θ
   • cot² θ ≡ 1 + cosec² θ
   • sec² θ ≡ tan² θ - 1
   Show answer

   cosec² θ ≡ 1 + cot² θ. Dividing sin² θ + cos² θ ≡ 1 by sin² θ gives 1 + cot² θ ≡ cosec² θ.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 3 scheme of work

Last updated 7 November 2026. Spotted a mistake? Tell us.