Skip to main content
MathidiaMathematics Academy
Hyperbolic functions
Lesson 32 of 75
Easy practice

Hyperbolic identities

Objective. Prove and apply cosh²x−sinh²x=1.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Hyperbolic identities mirror trigonometric ones but with sign differences arising from the exponential definitions; the key identity is cosh²x − sinh²x = 1, proved by substituting the exponential forms and expanding. Other identities follow similarly, such as sinh(2x) = 2 sinh x cosh x and cosh(2x) = cosh²x + sinh²x = 1 + 2sinh²x. These identities are used to simplify expressions and solve equations involving hyperbolic functions, exactly as Pythagorean and double-angle identities are used in trigonometry.

Worked example

Prove the identity cosh²x − sinh²x = 1 using the exponential definitions of cosh x and sinh x.

1
Illustrated card showing 1
  1. 1cosh x = (eˣ+e⁻ˣ)/2, so cosh²x = (e^(2x) + 2 + e^(−2x))/4
  2. 2sinh x = (eˣ−e⁻ˣ)/2, so sinh²x = (e^(2x) − 2 + e^(−2x))/4
  3. 3Subtract: cosh²x − sinh²x = [(e^(2x)+2+e^(−2x)) − (e^(2x)−2+e^(−2x))]/4
  4. 4Simplify numerator: 4/4 = 1

Answer: cosh²x − sinh²x = 1, proved

Concept mastery

Based on your past attempts at this lesson — the question types to practise again come first.

Complete a practice set to see your concept breakdown here.

Practice

Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.

Start practising

Common mistake. Students write cosh²x + sinh²x = 1, copying the trigonometric identity exactly. The hyperbolic identity has a minus sign: cosh²x − sinh²x = 1, which must be re-derived or memorised separately.

Your progress

Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.

0 of 5 steps complete

Your place saves automatically as you read.

My dashboard