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Review hyperbolic trigonometric identities

step inference rule input feed output step validity (as per SymPy)
1
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 6404535647
    \(\cosh x = \frac{\exp(x) + \exp(-x)}{2}\)
 no validation is available for declarations
2
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 1038566242
    \(\sinh x = \frac{\exp(x) - \exp(-x)}{2}\)
 no validation is available for declarations
3
  • 0000111253: multiply expr 1 by expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Multiply Eq.~\ref{eq:#1} by Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 1038566242
    \(\sinh x = \frac{\exp(x) - \exp(-x)}{2}\)
  1. 1038566242
    \(\sinh x = \frac{\exp(x) - \exp(-x)}{2}\)
  1. 6031385191
    \(\sinh^2 x = \left(\frac{\exp(x) - \exp(-x)}{2}\right)\left(\frac{\exp(x) - \exp(-x)}{2}\right)\)
 valid
4
  • 0000111253: multiply expr 1 by expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Multiply Eq.~\ref{eq:#1} by Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 6404535647
    \(\cosh x = \frac{\exp(x) + \exp(-x)}{2}\)
  1. 6404535647
    \(\cosh x = \frac{\exp(x) + \exp(-x)}{2}\)
  1. 8532702080
    \(\cosh^2 x = \left(\frac{\exp(x) + \exp(-x)}{2}\right)\left(\frac{\exp(x) + \exp(-x)}{2}\right)\)
 valid
5
  • 0000111222: subtract expr 1 from expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Subtract Eq.~\ref{eq:#1} from Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#2}.
  1. 8532702080
    \(\cosh^2 x = \left(\frac{\exp(x) + \exp(-x)}{2}\right)\left(\frac{\exp(x) + \exp(-x)}{2}\right)\)
  1. 6031385191
    \(\sinh^2 x = \left(\frac{\exp(x) - \exp(-x)}{2}\right)\left(\frac{\exp(x) - \exp(-x)}{2}\right)\)
  1. 8563535636
    \(\cosh^2 x - \sinh^2 x = \left(\frac{\exp(x) + \exp(-x)}{2}\right)\left(\frac{\exp(x) + \exp(-x)}{2}\right) - \left(\frac{\exp(x) - \exp(-x)}{2}\right)\left(\frac{\exp(x) - \exp(-x)}{2}\right)\)
 LHS diff is -2 RHS diff is -2
6
  • 0000111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 8563535636
    \(\cosh^2 x - \sinh^2 x = \left(\frac{\exp(x) + \exp(-x)}{2}\right)\left(\frac{\exp(x) + \exp(-x)}{2}\right) - \left(\frac{\exp(x) - \exp(-x)}{2}\right)\left(\frac{\exp(x) - \exp(-x)}{2}\right)\)
  1. 2762326680
    \(\cosh^2 x - \sinh^2 x = \frac{1}{4}\left( \exp(2x)+1+1+\exp(-2x) - \left(\exp(2x)-1-1+\exp(-2x)\right) \right)\)
 valid
7
  • 0000111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 2762326680
    \(\cosh^2 x - \sinh^2 x = \frac{1}{4}\left( \exp(2x)+1+1+\exp(-2x) - \left(\exp(2x)-1-1+\exp(-2x)\right) \right)\)
  1. 9413609246
    \(\cosh^2 x - \sinh^2 x = 1\)
 valid
8
  • 0000111341: declare final expression
  • number of inputs: 1; feeds: 0; outputs: 0
  • Eq.~\ref{eq:#1} is one of the final equations.
  1. 9413609246
    \(\cosh^2 x - \sinh^2 x = 1\)
 no validation is available for declarations
9
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 8747785338
    \(\cos(i x) = \cosh(x)\)
 no validation is available for declarations
10
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 8418527415
    \(\sin(i x) = i \sinh(x)\)
 no validation is available for declarations
11
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 2103023049
    \(\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)\)
 no validation is available for declarations
12
  • 0000111886: change variable X to Y
  • number of inputs: 1; feeds: 2; outputs: 1
  • Change variable $#1$ to $#2$ in Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}.
  1. 2103023049
    \(\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)\)
  1. 7159989263
    \(i x\)
  1. 6976493023
    \(x\)
  1. 4878728014
    \(\sin(i x) = \frac{1}{2i}\left(\exp(-x) - \exp(x) \right)\)
 LHS diff is sin(pdg0001464) - sin(pdg0001464*pdg0004621) RHS diff is 2*sinh(pdg0001464)/pdg0004621
13
  • 0000111355: LHS of expr 1 equals LHS of expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • LHS of Eq.~\ref{eq:#1} is equal to LHS of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 8418527415
    \(\sin(i x) = i \sinh(x)\)
  1. 4878728014
    \(\sin(i x) = \frac{1}{2i}\left(\exp(-x) - \exp(x) \right)\)
  1. 5323719091
    \(i \sinh x = \frac{1}{2i} \left( \exp(-x) - \exp(x) \right)\)
 valid
14
  • 0000111182: multiply both sides by
  • number of inputs: 1; feeds: 1; outputs: 1
  • Multiply both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.
  1. 5323719091
    \(i \sinh x = \frac{1}{2i} \left( \exp(-x) - \exp(x) \right)\)
  1. 9885190237
    \(i\)
  1. 1038566242
    \(\sinh x = \frac{\exp(x) - \exp(-x)}{2}\)
 LHS arithmetic error. Diff: (pdg0004621**2 - 1)*sinh(pdg0001464)
15
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 4585932229
    \(\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)\)
 no validation is available for declarations
16
  • 0000111886: change variable X to Y
  • number of inputs: 1; feeds: 2; outputs: 1
  • Change variable $#1$ to $#2$ in Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}.
  1. 4585932229
    \(\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)\)
  1. 1716984328
    \(i x\)
  1. 7453225570
    \(x\)
  1. 8651044341
    \(\cos(i x) = \frac{1}{2} \left( \exp(-x) + \exp(x) \right)\)
 LHS diff is cos(pdg0001464) - cos(pdg0001464*pdg0004621) RHS diff is 0
17
  • 0000111355: LHS of expr 1 equals LHS of expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • LHS of Eq.~\ref{eq:#1} is equal to LHS of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 8651044341
    \(\cos(i x) = \frac{1}{2} \left( \exp(-x) + \exp(x) \right)\)
  1. 8747785338
    \(\cos(i x) = \cosh(x)\)
  1. 6404535647
    \(\cosh x = \frac{\exp(x) + \exp(-x)}{2}\)
 valid
18
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 7731226616
    \({\rm sech}\ x = \frac{1}{\cosh x}\)
 no validation is available for declarations
19
  • 0000111556: substitute LHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 7731226616
    \({\rm sech}\ x = \frac{1}{\cosh x}\)
  1. 6404535647
    \(\cosh x = \frac{\exp(x) + \exp(-x)}{2}\)
  1. 4166155526
    \({\rm sech}\ x = \frac{2}{\exp(x)+\exp(-x)}\)
 LHS diff is cosh(pdg0001464) - sech(pdg0001464) RHS diff is (exp(4*pdg0001464) - 2*exp(2*pdg0001464) + 1)*exp(-pdg0001464)/(2*(exp(2*pdg0001464) + 1))
20
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 4872163189
    \(\tanh(x) = \frac{\sinh(x)}{\cosh(x)}\)
 no validation is available for declarations
21
  • 0000111556: substitute LHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 4872163189
    \(\tanh(x) = \frac{\sinh(x)}{\cosh(x)}\)
  1. 1038566242
    \(\sinh x = \frac{\exp(x) - \exp(-x)}{2}\)
  1. 2902772962
    \(\tanh(x) = \frac{\frac{1}{2}\left( \exp(x)-\exp(-x) \right)}{\cosh(x)}\)
 LHS diff is sinh(pdg0001464) - tanh(pdg0001464) RHS diff is (exp(2*pdg0001464) - exp(2*pdg0001464)/cosh(pdg0001464) - 1 + 1/cosh(pdg0001464))*exp(-pdg0001464)/2
22
  • 0000111556: substitute LHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 2902772962
    \(\tanh(x) = \frac{\frac{1}{2}\left( \exp(x)-\exp(-x) \right)}{\cosh(x)}\)
  1. 6404535647
    \(\cosh x = \frac{\exp(x) + \exp(-x)}{2}\)
  1. 5349669879
    \(\tanh(x) = \frac{ \exp(x)-\exp(-x)}{\exp(x)+\exp(-x)}\)
 LHS diff is cosh(pdg0001464) - tanh(pdg0001464) RHS diff is (exp(4*pdg0001464)/2 - exp(3*pdg0001464) + exp(2*pdg0001464) + exp(pdg0001464) + 1/2)*exp(-pdg0001464)/(exp(2*pdg0001464) + 1)
23
  • 0000111253: multiply expr 1 by expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Multiply Eq.~\ref{eq:#1} by Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 5349669879
    \(\tanh(x) = \frac{ \exp(x)-\exp(-x)}{\exp(x)+\exp(-x)}\)
  1. 5349669879
    \(\tanh(x) = \frac{ \exp(x)-\exp(-x)}{\exp(x)+\exp(-x)}\)
  1. 2121790783
    \(\tanh^2(x) = \frac{ \left(\exp(x)-\exp(-x)\right)^2}{\left(\exp(x)+\exp(-x)\right)^2}\)
 valid
24
  • 0000111253: multiply expr 1 by expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Multiply Eq.~\ref{eq:#1} by Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 4166155526
    \({\rm sech}\ x = \frac{2}{\exp(x)+\exp(-x)}\)
  1. 4166155526
    \({\rm sech}\ x = \frac{2}{\exp(x)+\exp(-x)}\)
  1. 3868998312
    \({\rm sech}^2\ x = \frac{4}{\left(\exp(x)+\exp(-x)\right)^2}\)
 valid
25
  • 0000111980: add expr 1 to expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Add Eq.~\ref{eq:#1} to Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#2}.
  1. 2121790783
    \(\tanh^2(x) = \frac{ \left(\exp(x)-\exp(-x)\right)^2}{\left(\exp(x)+\exp(-x)\right)^2}\)
  1. 3868998312
    \({\rm sech}^2\ x = \frac{4}{\left(\exp(x)+\exp(-x)\right)^2}\)
  1. 1128605625
    \({\rm sech}^2\ x + \tanh^2(x) = \frac{4}{\left(\exp(x)+\exp(-x)\right)^2} + \frac{\left(\exp(x)-\exp(-x)\right)^2}{\left(\exp(x)+\exp(-x)\right)^2}\)
 valid
26
  • 0000111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 1128605625
    \({\rm sech}^2\ x + \tanh^2(x) = \frac{4}{\left(\exp(x)+\exp(-x)\right)^2} + \frac{\left(\exp(x)-\exp(-x)\right)^2}{\left(\exp(x)+\exp(-x)\right)^2}\)
  1. 4830221561
    \({\rm sech}^2\ x + \tanh^2(x) = \frac{4+\left(\exp(2x)-1-1+\exp(-2x)\right)}{\left(\exp(x)+\exp(-x)\right)^2}\)
 valid
27
  • 0000111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 4830221561
    \({\rm sech}^2\ x + \tanh^2(x) = \frac{4+\left(\exp(2x)-1-1+\exp(-2x)\right)}{\left(\exp(x)+\exp(-x)\right)^2}\)
  1. 5866629429
    \({\rm sech}^2\ x + \tanh^2(x) = 1\)
 valid
28
  • 0000111341: declare final expression
  • number of inputs: 1; feeds: 0; outputs: 0
  • Eq.~\ref{eq:#1} is one of the final equations.
  1. 5866629429
    \({\rm sech}^2\ x + \tanh^2(x) = 1\)
 no validation is available for declarations

Symbols used in hyperbolic trigonometric identities

Steps and expressions for hyperbolic trigonometric identities

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