Math 2.0

statement proposition

cosh^2 - sinh^2 = 1

Statement

For every real xx, cosh⁡2x−sinh⁡2x=1.\cosh^2 x - \sinh^2 x = 1.

Remarks

The left side factors as (cosh⁡x−sinh⁡x)(cosh⁡x+sinh⁡x)=e−xex=1(\cosh x - \sinh x)(\cosh x + \sinh x) = e^{-x} e^{x} = 1. So the point (cosh⁡t,sinh⁡t)(\cosh t, \sinh t) lies on the right branch of the hyperbola x2−y2=1x^2 - y^2 = 1. Dividing by cosh⁡2x\cosh^2 x gives 1−tanh⁡2x=sech⁡2x1 - \tanh^2 x = \operatorname{sech}^2 x. Mathlib states the identity as Real.cosh_sq_sub_sinh_sq.

Explanations

L1 human

This is the hyperbolic version of cos⁡2x+sin⁡2x=1\cos^2 x + \sin^2 x = 1, with a minus sign. It follows directly from the definitions with exe^x and e−xe^{-x}.

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Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionGeometric sequence, definition, no evidence levelGeometric sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionSum of a finite geometric series, statement, no evidence levelSum of a finite geome…statementMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionSqueeze theorem, statement, no evidence levelSqueeze theoremstatementPartial sum, definition, no evidence levelPartial sumdefinitionPower series, definition, no evidence levelPower seriesdefinitionThe sequence r^n tends to zero for |r| < 1, example, no evidence levelThe sequence r^n tend…exampleRadius of convergence, definition, no evidence levelRadius of convergencedefinitionConvergent series, definition, no evidence levelConvergent seriesdefinitionA series of nonnegative terms converges iff its partial sums are bounded, statement, no evidence levelA series of nonnegati…statementAbsolute convergence, definition, no evidence levelAbsolute convergencedefinitionSum of a series, definition, no evidence levelSum of a seriesdefinitionComparison test, statement, no evidence levelComparison teststatementGeometric series, example, no evidence levelGeometric seriesexampleRatio test, statement, no evidence levelRatio teststatementRatio formula for the radius of convergence, statement, no evidence levelRatio formula for the…statementThe exponential series has infinite radius of convergence, statement, no evidence levelThe exponential serie…statementExponential function, definition, no evidence levelExponential functiondefinitionHyperbolic cosine, definition, no evidence levelHyperbolic cosinedefinitionHyperbolic sine, definition, no evidence levelHyperbolic sinedefinitioncosh^2 - sinh^2 = 1, statement, no evidence levelcosh^2 - sinh^2 = 1statement
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  1. Sequence definition
  2. Real sequence definition
  3. Bounded sequence definition
  4. Geometric sequence definition
  5. Limit of a sequence definition
  6. Monotone sequence definition
  7. Series definition
  8. Sum of a finite geometric series statement
  9. Monotone convergence theorem for sequences statement
  10. Convergent sequence definition
  11. Squeeze theorem statement
  12. Partial sum definition
  13. Power series definition
  14. The sequence r^n tends to zero for |r| < 1 example
  15. Radius of convergence definition
  16. Convergent series definition
  17. A series of nonnegative terms converges iff its partial sums are bounded statement
  18. Absolute convergence definition
  19. Sum of a series definition
  20. Comparison test statement
  21. Geometric series example
  22. Ratio test statement
  23. Ratio formula for the radius of convergence statement
  24. The exponential series has infinite radius of convergence statement
  25. Exponential function definition
  26. Hyperbolic cosine definition
  27. Hyperbolic sine definition
  28. cosh^2 - sinh^2 = 1 statement

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Sources

  1. Web https://en.wikipedia.org/wiki/Hyperbolic_functions, Useful relations

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