Hyperbolic identities

The key identity

cosh² x - sinh² x ≡ 1.

Proof: cosh² x = (e^2x + 2 + e^-2x)/4 and sinh² x = (e^2x - 2 + e^-2x)/4. Subtracting leaves 4/4 = 1.

Dividing by cosh² x gives 1 - tanh² x ≡ sech² x. Dividing by sinh² x gives coth² x - 1 ≡ cosech² x.

Double angle and compound angle

• sinh 2x ≡ 2 sinh x cosh x
• cosh 2x ≡ cosh² x + sinh² x ≡ 2cosh² x - 1 ≡ 1 + 2sinh² x
• sinh(A ± B) ≡ sinh A cosh B ± cosh A sinh B
• cosh(A ± B) ≡ cosh A cosh B ± sinh A sinh B

Each can be proved by writing everything as exponentials and simplifying.

Spot the sign change

Compare with trigonometry: cos² x + sin² x ≡ 1 becomes cosh² x - sinh² x ≡ 1, and cos 2x ≡ 1 - 2sin² x becomes cosh 2x ≡ 1 + 2sinh² x. The sign changes wherever a product of two sines appears (Osborn's rule is a useful check, but in the exam you prove identities from the definitions).

Using the identities

If you know one hyperbolic value you can find the others. Remember cosh x is always positive, while sinh x and tanh x take the sign of x.

Worked example

Given sinh x = 3/4, find cosh x and tanh x.

1. cosh² x = 1 + sinh² x = 1 + 9/16 = 25/16
2. cosh x > 0, so cosh x = 5/4
3. tanh x = sinh x / cosh x = (3/4)/(5/4)

Answer: cosh x = 5/4 and tanh x = 3/5

Check yourself

1. Given sinh x = 3/4, find cosh x. Give your answer as a fraction in its simplest form.
   Show answer

   5/4. cosh² x = 1 + 9/16 = 25/16 and cosh x is positive, so cosh x = 5/4.

2. Given tanh x = 1/2, find sech² x. Give your answer as a fraction in its simplest form.
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   3/4. sech² x = 1 - tanh² x = 1 - 1/4 = 3/4.

3. Given sinh x = 2, find cosh 2x.
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   9. cosh 2x = 1 + 2sinh² x = 1 + 2(4) = 9.

4. Given cosh x = 13/12 and x > 0, find tanh x. Give your answer as a fraction in its simplest form.
   Show answer

   5/13. sinh² x = 169/144 - 1 = 25/144, so sinh x = 5/12 (positive because x > 0). tanh x = (5/12)/(13/12) = 5/13.

5. Which identity is correct?
   • cosh 2x ≡ 1 - 2sinh² x
   • cosh 2x ≡ 2cosh² x - 1
   • cosh 2x ≡ cosh² x - sinh² x
   • cosh 2x ≡ 2 sinh x cosh x
   Show answer

   cosh 2x ≡ 2cosh² x - 1. cosh 2x = cosh² x + sinh² x = cosh² x + (cosh² x - 1) = 2cosh² x - 1.

6. Which identity is the hyperbolic version of cos² x + sin² x ≡ 1?
   • cosh² x + sinh² x ≡ 1
   • sinh² x - cosh² x ≡ 1
   • cosh² x + sinh² x ≡ cosh x
   • cosh² x - sinh² x ≡ 1
   Show answer

   cosh² x - sinh² x ≡ 1. Expanding the exponentials gives cosh² x - sinh² x = 1.

7. Which expansion is correct?
   • cosh(A + B) ≡ cosh A cosh B - sinh A sinh B
   • cosh(A + B) ≡ cosh A cosh B + sinh A sinh B
   • cosh(A + B) ≡ sinh A cosh B + cosh A sinh B
   • cosh(A + B) ≡ cosh A + cosh B
   Show answer

   cosh(A + B) ≡ cosh A cosh B + sinh A sinh B. The product of two sinhs carries the sign change: cos(A + B) has - sin A sin B, so cosh(A + B) has + sinh A sinh B.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics, Further Mathematics and Pure Mathematics specification (issue 3), Unit FP3

Last updated 8 November 2026. Spotted a mistake? Tell us.