Hyperbolic Identities
cosh²x − sinh²x = 1.
Key idea
cosh²x − sinh²x = 1.
In depth
cosh²x − sinh²x = 1, mirroring the sign-changed Pythagorean identity. Also sinh 2x = 2 sinh x cosh x. Osborn’s rule: replace trig identities by hyperbolic ones, changing the sign of any product of two sinh terms.
Worked example
cosh²(2) − sinh²(2) = 1.
Real-world uses
- Simplifying formulas for hanging cables and special relativity calculations
Common mistake
Writing cosh² + sinh² = 1.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Hyperbolic Identities?
cosh²x − sinh²x = 1. It is a topic at the expert level in Trigonometry.
Where is Hyperbolic Identities used in real life?
Real-world examples: Simplifying formulas for hanging cables and special relativity calculations.
What should I learn before Hyperbolic Identities?
Make sure you are comfortable with Hyperbolic Functions, the previous topic in Trigonometry.
What is a common mistake with Hyperbolic Identities?
Writing cosh² + sinh² = 1.
How can I practise Hyperbolic Identities?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Hyperbolic Identities?
Move on to Inverse Hyperbolic Functions next.
Related expert topics in Trigonometry
- Half Angle Formulas – Trig of θ/2.
- Power Reducing Formulas – Reduce squares to first powers of cos 2θ.
- Product-to-Sum Formulas – Convert products into sums.
- Sum-to-Product Formulas – Convert sums into products.
- Triple Angle Formulas – sin3θ and cos3θ in terms of θ.
- Trig and Inverse Trig Compositions – Evaluate expressions like sin(arccos x).