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Review hyperbolic trigonometric identities

abstract:
reference: http://www.physics.miami.edu/~nearing/mathmethods/mathematical_methods-one.pdf

step	inference rule	input	feed	output	validity (as per SymPy)
11	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			2103023049 $\sin(x)=\frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)$	no validation is available for declarations
26	ID: 111457; simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	1128605625 ${\rm sech}^2\ x + \tanh^2(x)=\frac{4}{\left(\exp(x)+\exp(-x)\right)^2} + \frac{\left(\exp(x)-\exp(-x)\right)^2}{\left(\exp(x)+\exp(-x)\right)^2}$		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
25	ID: 111980; 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}.	2121790783 $\tanh^2(x)=\frac{ \left(\exp(x)-\exp(-x)\right)^2}{\left(\exp(x)+\exp(-x)\right)^2}$ 3868998312 ${\rm sech}^2\ x=\frac{4}{\left(\exp(x)+\exp(-x)\right)^2}$		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
15	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			4585932229 $\cos(x)=\frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)$	no validation is available for declarations
19	ID: 111556; 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}.	7731226616 ${\rm sech}\ x=\frac{1}{\cosh x}$ 6404535647 $\cosh x=\frac{\exp(x) + \exp(-x)}{2}$		4166155526 ${\rm sech}\ x=\frac{2}{\exp(x)+\exp(-x)}$	LHS diff is cosh(pdg0001464) - sech(pdg0001464) RHS diff is (exp(4pdg0001464) - 2exp(2pdg0001464) + 1)exp(-pdg0001464)/(2(exp(2pdg0001464) + 1))
21	ID: 111556; 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}.	1038566242 $\sinh x=\frac{\exp(x) - \exp(-x)}{2}$ 4872163189 $\tanh(x)=\frac{\sinh(x)}{\cosh(x)}$		2902772962 $\tanh(x)=\frac{\frac{1}{2}\left( \exp(x)-\exp(-x) \right)}{\cosh(x)}$	valid
27	ID: 111457; simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	4830221561 ${\rm sech}^2\ x + \tanh^2(x)=\frac{4+\left(\exp(2x)-1-1+\exp(-2x)\right)}{\left(\exp(x)+\exp(-x)\right)^2}$		5866629429 ${\rm sech}^2\ x + \tanh^2(x)=1$	valid
2	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			1038566242 $\sinh x=\frac{\exp(x) - \exp(-x)}{2}$	no validation is available for declarations
12	ID: 111886; 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}.	2103023049 $\sin(x)=\frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)$	6976493023 $x$ 7159989263 $i x$	4878728014 $\sin(i x)=\frac{1}{2i}\left(\exp(-x) - \exp(x) \right)$	LHS diff is 0 RHS diff is exp(pdg0001464)/(2pdg0004621) + exp(pdg0001464pdg0004621*2)/(2pdg0004621) - exp(-pdg0001464pdg00046212)/(2pdg0004621) - exp(-pdg0001464)/(2*pdg0004621)
24	ID: 111253; 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}.	4166155526 ${\rm sech}\ x=\frac{2}{\exp(x)+\exp(-x)}$ 4166155526 ${\rm sech}\ x=\frac{2}{\exp(x)+\exp(-x)}$		3868998312 ${\rm sech}^2\ x=\frac{4}{\left(\exp(x)+\exp(-x)\right)^2}$	valid
28	ID: 111341; declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\\ref{eq:#1} is one of the final equations.	5866629429 ${\rm sech}^2\ x + \tanh^2(x)=1$			no validation is available for declarations
14	ID: 111182; 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}.	5323719091 $i \sinh x=\frac{1}{2i} \left( \exp(-x) - \exp(x) \right)$	9885190237 $i$	1038566242 $\sinh x=\frac{\exp(x) - \exp(-x)}{2}$	LHS arithmetic error. Diff: (pdg0004621*2 - 1)sinh(pdg0001464)
7	ID: 111457; simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	2762326680 $\cosh^2 x - \sinh^2 x=\frac{1}{4}\left( \exp(2x)+1+1+\exp(-2x) - \left(\exp(2x)-1-1+\exp(-2x)\right) \right)$		9413609246 $\cosh^2 x - \sinh^2 x=1$	valid
3	ID: 111253; 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}.	1038566242 $\sinh x=\frac{\exp(x) - \exp(-x)}{2}$ 1038566242 $\sinh x=\frac{\exp(x) - \exp(-x)}{2}$		6031385191 $\sinh^2 x=\left(\frac{\exp(x) - \exp(-x)}{2}\right)\left(\frac{\exp(x) - \exp(-x)}{2}\right)$	valid
16	ID: 111886; 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}.	4585932229 $\cos(x)=\frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)$	1716984328 $i x$ 7453225570 $x$	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
8	ID: 111341; declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\\ref{eq:#1} is one of the final equations.	9413609246 $\cosh^2 x - \sinh^2 x=1$			no validation is available for declarations
10	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			8418527415 $\sin(i x)=i \sinh(x)$	no validation is available for declarations
22	ID: 111556; 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}.	2902772962 $\tanh(x)=\frac{\frac{1}{2}\left( \exp(x)-\exp(-x) \right)}{\cosh(x)}$ 6404535647 $\cosh x=\frac{\exp(x) + \exp(-x)}{2}$		5349669879 $\tanh(x)=\frac{ \exp(x)-\exp(-x)}{\exp(x)+\exp(-x)}$	LHS diff is cosh(pdg0001464) - tanh(pdg0001464) RHS diff is (exp(4pdg0001464)/2 - exp(3pdg0001464) + exp(2pdg0001464) + exp(pdg0001464) + 1/2)exp(-pdg0001464)/(exp(2*pdg0001464) + 1)
13	ID: 111355; 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}.	4878728014 $\sin(i x)=\frac{1}{2i}\left(\exp(-x) - \exp(x) \right)$ 8418527415 $\sin(i x)=i \sinh(x)$		5323719091 $i \sinh x=\frac{1}{2i} \left( \exp(-x) - \exp(x) \right)$	input diff is 0 diff is -pdg0004621sinh(pdg0001464) - exp(pdg0001464)/(2pdg0004621) + exp(-pdg0001464)/(2pdg0004621) diff is pdg0004621sinh(pdg0001464) + exp(pdg0001464)/(2pdg0004621) - exp(-pdg0001464)/(2pdg0004621)
20	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			4872163189 $\tanh(x)=\frac{\sinh(x)}{\cosh(x)}$	no validation is available for declarations
17	ID: 111355; 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}.	8747785338 $\cos(i x)=\cosh(x)$ 8651044341 $\cos(i x)=\frac{1}{2} \left( \exp(-x) + \exp(x) \right)$		6404535647 $\cosh x=\frac{\exp(x) + \exp(-x)}{2}$	valid
18	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			7731226616 ${\rm sech}\ x=\frac{1}{\cosh x}$	no validation is available for declarations
23	ID: 111253; 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}.	5349669879 $\tanh(x)=\frac{ \exp(x)-\exp(-x)}{\exp(x)+\exp(-x)}$ 5349669879 $\tanh(x)=\frac{ \exp(x)-\exp(-x)}{\exp(x)+\exp(-x)}$		2121790783 $\tanh^2(x)=\frac{ \left(\exp(x)-\exp(-x)\right)^2}{\left(\exp(x)+\exp(-x)\right)^2}$	valid
6	ID: 111457; simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	8563535636 $\cosh^2 x - \sinh^2 x=\left(\frac{\exp(x) + \exp(-x)}{2}\right)\left(\frac{\exp(x) + \exp(-x)}{2}\right) - \left(\frac{\exp(x) - \exp(-x)}{2}\right)\left(\frac{\exp(x) - \exp(-x)}{2}\right)$		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
9	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			8747785338 $\cos(i x)=\cosh(x)$	no validation is available for declarations
5	ID: 111222; 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}.	6031385191 $\sinh^2 x=\left(\frac{\exp(x) - \exp(-x)}{2}\right)\left(\frac{\exp(x) - \exp(-x)}{2}\right)$ 8532702080 $\cosh^2 x=\left(\frac{\exp(x) + \exp(-x)}{2}\right)\left(\frac{\exp(x) + \exp(-x)}{2}\right)$		8563535636 $\cosh^2 x - \sinh^2 x=\left(\frac{\exp(x) + \exp(-x)}{2}\right)\left(\frac{\exp(x) + \exp(-x)}{2}\right) - \left(\frac{\exp(x) - \exp(-x)}{2}\right)\left(\frac{\exp(x) - \exp(-x)}{2}\right)$	valid
4	ID: 111253; 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}.	6404535647 $\cosh x=\frac{\exp(x) + \exp(-x)}{2}$ 6404535647 $\cosh x=\frac{\exp(x) + \exp(-x)}{2}$		8532702080 $\cosh^2 x=\left(\frac{\exp(x) + \exp(-x)}{2}\right)\left(\frac{\exp(x) + \exp(-x)}{2}\right)$	valid
1	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			6404535647 $\cosh x=\frac{\exp(x) + \exp(-x)}{2}$	no validation is available for declarations

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0.002 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule2954526
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0.002 seconds for compute/get_dict_of_steps_in_derivation: step_has_inference_rule7421979
0.002 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_INPUT7421979
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0.002 seconds for compute/get_dict_of_steps_in_derivation: step_id_has_expressions, HAS_OUTPUT7421979

step	inference rule	input	feed	output	validity (as per SymPy)
11	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			2103023049 \(\sin(x)=\frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)\)	no validation is available for declarations
26	ID: 111457; simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	1128605625 \({\rm sech}^2\ x + \tanh^2(x)=\frac{4}{\left(\exp(x)+\exp(-x)\right)^2} + \frac{\left(\exp(x)-\exp(-x)\right)^2}{\left(\exp(x)+\exp(-x)\right)^2}\)		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
25	ID: 111980; 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}.	2121790783 \(\tanh^2(x)=\frac{ \left(\exp(x)-\exp(-x)\right)^2}{\left(\exp(x)+\exp(-x)\right)^2}\) 3868998312 \({\rm sech}^2\ x=\frac{4}{\left(\exp(x)+\exp(-x)\right)^2}\)		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
15	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			4585932229 \(\cos(x)=\frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)\)	no validation is available for declarations
19	ID: 111556; 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}.	7731226616 \({\rm sech}\ x=\frac{1}{\cosh x}\) 6404535647 \(\cosh x=\frac{\exp(x) + \exp(-x)}{2}\)		4166155526 \({\rm sech}\ x=\frac{2}{\exp(x)+\exp(-x)}\)	LHS diff is cosh(pdg0001464) - sech(pdg0001464) RHS diff is (exp(4pdg0001464) - 2exp(2pdg0001464) + 1)exp(-pdg0001464)/(2(exp(2pdg0001464) + 1))
21	ID: 111556; 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}.	1038566242 \(\sinh x=\frac{\exp(x) - \exp(-x)}{2}\) 4872163189 \(\tanh(x)=\frac{\sinh(x)}{\cosh(x)}\)		2902772962 \(\tanh(x)=\frac{\frac{1}{2}\left( \exp(x)-\exp(-x) \right)}{\cosh(x)}\)	valid
27	ID: 111457; simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	4830221561 \({\rm sech}^2\ x + \tanh^2(x)=\frac{4+\left(\exp(2x)-1-1+\exp(-2x)\right)}{\left(\exp(x)+\exp(-x)\right)^2}\)		5866629429 \({\rm sech}^2\ x + \tanh^2(x)=1\)	valid
2	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			1038566242 \(\sinh x=\frac{\exp(x) - \exp(-x)}{2}\)	no validation is available for declarations
12	ID: 111886; 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}.	2103023049 \(\sin(x)=\frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)\)	6976493023 \(x\) 7159989263 \(i x\)	4878728014 \(\sin(i x)=\frac{1}{2i}\left(\exp(-x) - \exp(x) \right)\)	LHS diff is 0 RHS diff is exp(pdg0001464)/(2pdg0004621) + exp(pdg0001464pdg0004621*2)/(2pdg0004621) - exp(-pdg0001464pdg00046212)/(2pdg0004621) - exp(-pdg0001464)/(2*pdg0004621)
24	ID: 111253; 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}.	4166155526 \({\rm sech}\ x=\frac{2}{\exp(x)+\exp(-x)}\) 4166155526 \({\rm sech}\ x=\frac{2}{\exp(x)+\exp(-x)}\)		3868998312 \({\rm sech}^2\ x=\frac{4}{\left(\exp(x)+\exp(-x)\right)^2}\)	valid
28	ID: 111341; declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\\ref{eq:#1} is one of the final equations.	5866629429 \({\rm sech}^2\ x + \tanh^2(x)=1\)			no validation is available for declarations
14	ID: 111182; 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}.	5323719091 \(i \sinh x=\frac{1}{2i} \left( \exp(-x) - \exp(x) \right)\)	9885190237 \(i\)	1038566242 \(\sinh x=\frac{\exp(x) - \exp(-x)}{2}\)	LHS arithmetic error. Diff: (pdg0004621*2 - 1)sinh(pdg0001464)
7	ID: 111457; simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	2762326680 \(\cosh^2 x - \sinh^2 x=\frac{1}{4}\left( \exp(2x)+1+1+\exp(-2x) - \left(\exp(2x)-1-1+\exp(-2x)\right) \right)\)		9413609246 \(\cosh^2 x - \sinh^2 x=1\)	valid
3	ID: 111253; 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}.	1038566242 \(\sinh x=\frac{\exp(x) - \exp(-x)}{2}\) 1038566242 \(\sinh x=\frac{\exp(x) - \exp(-x)}{2}\)		6031385191 \(\sinh^2 x=\left(\frac{\exp(x) - \exp(-x)}{2}\right)\left(\frac{\exp(x) - \exp(-x)}{2}\right)\)	valid
16	ID: 111886; 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}.	4585932229 \(\cos(x)=\frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)\)	1716984328 \(i x\) 7453225570 \(x\)	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
8	ID: 111341; declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\\ref{eq:#1} is one of the final equations.	9413609246 \(\cosh^2 x - \sinh^2 x=1\)			no validation is available for declarations
10	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			8418527415 \(\sin(i x)=i \sinh(x)\)	no validation is available for declarations
22	ID: 111556; 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}.	2902772962 \(\tanh(x)=\frac{\frac{1}{2}\left( \exp(x)-\exp(-x) \right)}{\cosh(x)}\) 6404535647 \(\cosh x=\frac{\exp(x) + \exp(-x)}{2}\)		5349669879 \(\tanh(x)=\frac{ \exp(x)-\exp(-x)}{\exp(x)+\exp(-x)}\)	LHS diff is cosh(pdg0001464) - tanh(pdg0001464) RHS diff is (exp(4pdg0001464)/2 - exp(3pdg0001464) + exp(2pdg0001464) + exp(pdg0001464) + 1/2)exp(-pdg0001464)/(exp(2*pdg0001464) + 1)
13	ID: 111355; 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}.	4878728014 \(\sin(i x)=\frac{1}{2i}\left(\exp(-x) - \exp(x) \right)\) 8418527415 \(\sin(i x)=i \sinh(x)\)		5323719091 \(i \sinh x=\frac{1}{2i} \left( \exp(-x) - \exp(x) \right)\)	input diff is 0 diff is -pdg0004621sinh(pdg0001464) - exp(pdg0001464)/(2pdg0004621) + exp(-pdg0001464)/(2pdg0004621) diff is pdg0004621sinh(pdg0001464) + exp(pdg0001464)/(2pdg0004621) - exp(-pdg0001464)/(2pdg0004621)
20	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			4872163189 \(\tanh(x)=\frac{\sinh(x)}{\cosh(x)}\)	no validation is available for declarations
17	ID: 111355; 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}.	8747785338 \(\cos(i x)=\cosh(x)\) 8651044341 \(\cos(i x)=\frac{1}{2} \left( \exp(-x) + \exp(x) \right)\)		6404535647 \(\cosh x=\frac{\exp(x) + \exp(-x)}{2}\)	valid
18	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			7731226616 \({\rm sech}\ x=\frac{1}{\cosh x}\)	no validation is available for declarations
23	ID: 111253; 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}.	5349669879 \(\tanh(x)=\frac{ \exp(x)-\exp(-x)}{\exp(x)+\exp(-x)}\) 5349669879 \(\tanh(x)=\frac{ \exp(x)-\exp(-x)}{\exp(x)+\exp(-x)}\)		2121790783 \(\tanh^2(x)=\frac{ \left(\exp(x)-\exp(-x)\right)^2}{\left(\exp(x)+\exp(-x)\right)^2}\)	valid
6	ID: 111457; simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.	8563535636 \(\cosh^2 x - \sinh^2 x=\left(\frac{\exp(x) + \exp(-x)}{2}\right)\left(\frac{\exp(x) + \exp(-x)}{2}\right) - \left(\frac{\exp(x) - \exp(-x)}{2}\right)\left(\frac{\exp(x) - \exp(-x)}{2}\right)\)		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
9	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			8747785338 \(\cos(i x)=\cosh(x)\)	no validation is available for declarations
5	ID: 111222; 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}.	6031385191 \(\sinh^2 x=\left(\frac{\exp(x) - \exp(-x)}{2}\right)\left(\frac{\exp(x) - \exp(-x)}{2}\right)\) 8532702080 \(\cosh^2 x=\left(\frac{\exp(x) + \exp(-x)}{2}\right)\left(\frac{\exp(x) + \exp(-x)}{2}\right)\)		8563535636 \(\cosh^2 x - \sinh^2 x=\left(\frac{\exp(x) + \exp(-x)}{2}\right)\left(\frac{\exp(x) + \exp(-x)}{2}\right) - \left(\frac{\exp(x) - \exp(-x)}{2}\right)\left(\frac{\exp(x) - \exp(-x)}{2}\right)\)	valid
4	ID: 111253; 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}.	6404535647 \(\cosh x=\frac{\exp(x) + \exp(-x)}{2}\) 6404535647 \(\cosh x=\frac{\exp(x) + \exp(-x)}{2}\)		8532702080 \(\cosh^2 x=\left(\frac{\exp(x) + \exp(-x)}{2}\right)\left(\frac{\exp(x) + \exp(-x)}{2}\right)\)	valid
1	ID: 111981; declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\\ref{eq:#1} is an initial equation.			6404535647 \(\cosh x=\frac{\exp(x) + \exp(-x)}{2}\)	no validation is available for declarations