Math 2.0

Special functions Proof

Proof of the addition formulas for hyperbolic functions

Builds on Hyperbolic sineHyperbolic cosineExponential functionExponential of a sum

Proof

We expand the right sides with the definitions and compare.

  1. Let xx and yy be real numbers. Write et=exp⁡(t)e^t = \exp(t) for the exponential function.
  2. Let p=exeyp = e^x e^y, q=exe−yq = e^x e^{-y}, r=e−xeyr = e^{-x} e^y and s=e−xe−ys = e^{-x} e^{-y}.
  3. By the exponential of a sum, p=ex+yp = e^{x + y} and s=e−(x+y)s = e^{-(x + y)}.
  4. By the definitions of the hyperbolic sine and the hyperbolic cosine, sinh⁡t=et−e−t2\sinh t = \frac{e^t - e^{-t}}{2} and cosh⁡t=et+e−t2\cosh t = \frac{e^t + e^{-t}}{2} for every real tt.
  5. By step 4, 4sinh⁡xcosh⁡y=(ex−e−x)(ey+e−y)=p+q−r−s4 \sinh x \cosh y = (e^x - e^{-x})(e^y + e^{-y}) = p + q - r - s.
  6. By step 4, 4cosh⁡xsinh⁡y=(ex+e−x)(ey−e−y)=p−q+r−s4 \cosh x \sinh y = (e^x + e^{-x})(e^y - e^{-y}) = p - q + r - s.
  7. Adding steps 5 and 6 gives 4(sinh⁡xcosh⁡y+cosh⁡xsinh⁡y)=2p−2s4(\sinh x \cosh y + \cosh x \sinh y) = 2p - 2s.
  8. By steps 3 and 4, 2p−2s=2(ex+y−e−(x+y))=4sinh⁡(x+y)2p - 2s = 2(e^{x+y} - e^{-(x+y)}) = 4 \sinh(x + y). With step 7 this proves sinh⁡(x+y)=sinh⁡xcosh⁡y+cosh⁡xsinh⁡y\sinh(x + y) = \sinh x \cosh y + \cosh x \sinh y.
  9. By step 4, 4cosh⁡xcosh⁡y=(ex+e−x)(ey+e−y)=p+q+r+s4 \cosh x \cosh y = (e^x + e^{-x})(e^y + e^{-y}) = p + q + r + s.
  10. By step 4, 4sinh⁡xsinh⁡y=(ex−e−x)(ey−e−y)=p−q−r+s4 \sinh x \sinh y = (e^x - e^{-x})(e^y - e^{-y}) = p - q - r + s.
  11. Adding steps 9 and 10 gives 4(cosh⁡xcosh⁡y+sinh⁡xsinh⁡y)=2p+2s4(\cosh x \cosh y + \sinh x \sinh y) = 2p + 2s.
  12. By steps 3 and 4, 2p+2s=2(ex+y+e−(x+y))=4cosh⁡(x+y)2p + 2s = 2(e^{x+y} + e^{-(x+y)}) = 4 \cosh(x + y). With step 11 this proves cosh⁡(x+y)=cosh⁡xcosh⁡y+sinh⁡xsinh⁡y\cosh(x + y) = \cosh x \cosh y + \sinh x \sinh y.

Idea

Each product of two hyperbolic functions expands into four products of exponentials. In the right combination, the mixed terms ex−ye^{x - y} and ey−xe^{y - x} cancel. The surviving terms ex+ye^{x+y} and e−(x+y)e^{-(x+y)} assemble into the hyperbolic function of x+yx + y.

Reading path

To understand this page from the ground up, read these 161 pages in order. The foundations come first. The path takes the proofs Proof of the Weierstrass M-test, Proof of the limit of a sum of sequences, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof of the commutativity of addition by induction, Proof that absolutely convergent series converge, Proof of the sum of a finite geometric series, Proof that convergent sequences are Cauchy, Proof of the Cauchy criterion for series, Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem, Proof that the exponential series has infinite radius of convergence, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the double negation law by a truth table, Proof that power series converge uniformly on smaller intervals inside the radius of convergence, Proof of strong induction from ordinary induction, Proof of the limit of a product of sequences, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of Pascal's rule from the recursive definition, Proof that Cauchy sequences are bounded, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof of the comparison test, Proof that limits preserve non-strict inequalities, Proof of the factorial formula for binomial coefficients by induction, Proof of the binomial theorem by induction, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of the Archimedean property in an Archimedean ordered field, Proof of the exponential of a sum by the Cauchy product, Proof that a decreasing sequence bounded below converges, Proof that the reals are Archimedean from the completeness axiom, Proof of the ratio test, Proof that convergent sequences are bounded, Proof of the monotone convergence theorem for sequences, Proof of the ratio formula for the radius of convergence, Proof that a series of nonnegative terms converges iff its partial sums are bounded, Proof that the Cauchy product of absolutely convergent series converges absolutely to the product.

  1. Binomial coefficient definition
  2. Construction of the reals from Cauchy sequences concept
  3. Definition by recursion concept
  4. Element definition
  5. Factorial definition
  6. Group concept
  7. Integers object
  8. Mathematical induction concept
  9. Natural numbers object
  10. Ordered field definition
  11. Peano axioms concept
  12. Proposition concept
  13. Rational numbers object
  14. Real numbers object
  15. Sequence definition
  16. Set bounded above definition
  17. Proof of Pascal's rule from the recursive definition proof
  18. Set definition
  19. Ring definition
  20. Ring of integers statement
  21. Archimedean ordered field definition
  22. Successor function definition
  23. Conjunction definition
  24. Disjunction definition
  25. Implication definition
  26. Negation definition
  27. Predicate definition
  28. The reals form a field statement
  29. Real sequence definition
  30. Subsequence definition
  31. Infimum definition
  32. Supremum definition
  33. Pascal's rule statement
  34. Ordered pair definition
  35. Commutative ring definition
  36. Addition of natural numbers definition
  37. Proof of the Archimedean property in an Archimedean ordered field proof
  38. Biconditional definition
  39. The rationals form a field statement
  40. The successor function is injective statement
  41. Truth table concept
  42. Zero is not a successor statement
  43. Modus ponens concept
  44. Universal quantifier definition
  45. Bounded sequence definition
  46. Cauchy sequence definition
  47. Geometric sequence definition
  48. Limit of a sequence definition
  49. Monotone sequence definition
  50. Series definition
  51. Completeness axiom statement
  52. Cartesian product definition
  53. Proof of the factorial formula for binomial coefficients by induction proof
  54. Archimedean property statement
  55. Proof of the commutativity of addition by induction proof
  56. Order on the natural numbers definition
  57. Subset definition
  58. The rationals form an ordered field statement
  59. Propositional logic concept
  60. Mathematical proof concept
  61. Proof of the sum of a finite geometric series proof
  62. Convergent sequence definition
  63. Pointwise convergence definition
  64. Cauchy product definition
  65. Partial sum definition
  66. Power series definition
  67. Factorial formula for binomial coefficients statement
  68. Binary relation definition
  69. Order on the reals definition
  70. Commutativity of addition statement
  71. Proof of strong induction from ordinary induction proof
  72. Tautology definition
  73. Uniform convergence definition
  74. Sum of a finite geometric series statement
  75. Radius of convergence definition
  76. Convergent series definition
  77. Proof of the binomial theorem by induction proof
  78. Antisymmetric relation definition
  79. Reflexive relation definition
  80. Transitive relation definition
  81. Strong induction statement
  82. Logical equivalence definition
  83. Uniform convergence on a set definition
  84. Sum of a series definition
  85. Series of functions definition
  86. Absolute convergence definition
  87. Binomial theorem statement
  88. Preorder definition
  89. Proof of the well-ordering principle from strong induction proof
  90. Proof of the double negation law by a truth table proof
  91. Partial order definition
  92. Well-ordering principle statement
  93. Double negation law statement
  94. Total order definition
  95. Proof by contradiction concept
  96. Proof that every sequence has a monotone subsequence, by peaks proof
  97. Proof that the reals form an ordered field, with the Cauchy sequence construction proof
  98. Every sequence has a monotone subsequence statement
  99. The reals form an ordered field statement
  100. Absolute value definition
  101. Proof of the approximation property of the supremum proof
  102. Proof of the triangle inequality for real numbers proof
  103. Proof that the absolute value is multiplicative proof
  104. Proof of the epsilon-N definition of the limit of a sequence proof
  105. Approximation property of the supremum statement
  106. Triangle inequality statement
  107. Absolute value is multiplicative statement
  108. Epsilon-N definition of the limit of a sequence statement
  109. Proof that the reals are Archimedean from the completeness axiom proof
  110. Proof of the limit of a sum of sequences proof
  111. Proof of the squeeze theorem for sequences proof
  112. Proof that limits preserve non-strict inequalities proof
  113. Proof that convergent sequences are bounded proof
  114. Proof that convergent sequences are Cauchy proof
  115. The reals are Archimedean statement
  116. Proof that Cauchy sequences are bounded proof
  117. Proof of the monotone convergence theorem for sequences proof
  118. Limit of a sum of sequences statement
  119. Limits preserve non-strict inequalities statement
  120. Convergent sequences are bounded statement
  121. Proof of the ratio formula for the radius of convergence proof
  122. Squeeze theorem statement
  123. Convergent sequences are Cauchy statement
  124. Cauchy sequences are bounded statement
  125. Monotone convergence theorem for sequences statement
  126. Proof of the limit of a product of sequences proof
  127. Proof that a series of nonnegative terms converges iff its partial sums are bounded proof
  128. Proof that a decreasing sequence bounded below converges proof
  129. The sequence r^n tends to zero for |r| < 1 example
  130. Decreasing sequence bounded below converges statement
  131. Limit of a product of sequences statement
  132. A series of nonnegative terms converges iff its partial sums are bounded statement
  133. Geometric series example
  134. Proof that the Cauchy product of absolutely convergent series converges absolutely to the product proof
  135. Proof of the ratio test proof
  136. Proof of the comparison test proof
  137. Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
  138. Comparison test statement
  139. Cauchy product of absolutely convergent series statement
  140. Bolzano–Weierstrass theorem statement
  141. Ratio test statement
  142. Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem proof
  143. Ratio formula for the radius of convergence statement
  144. Cauchy criterion for sequences statement
  145. Uniformly Cauchy sequence of functions definition
  146. Proof of the Cauchy criterion for series proof
  147. Cauchy criterion for series statement
  148. Proof that absolutely convergent series converge proof
  149. Absolutely convergent series converge statement
  150. Proof of the Weierstrass M-test proof
  151. Weierstrass M-test statement
  152. Proof that power series converge uniformly on smaller intervals inside the radius of convergence proof
  153. Power series converge uniformly on smaller discs inside the radius of convergence statement
  154. Proof that the exponential series has infinite radius of convergence proof
  155. The exponential series has infinite radius of convergence statement
  156. Exponential function definition
  157. Hyperbolic cosine definition
  158. Hyperbolic sine definition
  159. Proof of the exponential of a sum by the Cauchy product proof
  160. Exponential of a sum statement
  161. Proof of the addition formulas for hyperbolic functions proof

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Hyperbolic_functions, Sums and differences of arguments
About this page
TagYDQ6P00V89
URIm2://YDQ6P00V89
Kindproof
WritingW0 stub
MSC33B10
ProvesAddition formulas for hyperbolic functions
UsesHyperbolic sine, Hyperbolic cosine, Exponential function, Exponential of a sum
Changed4 November 2026

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