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Review Euler equation: trigonometric relations

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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.			4938429483 $\exp(i x) = \cos(x)+i \sin(x)$	no validation is available for declarations
2	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}.	4938429483 $\exp(i x) = \cos(x)+i \sin(x)$	0003949052 $-x$ 0002393922 $x$	2394853829 $\exp(-i x) = \cos(-x)+i \sin(-x)$	LHS diff is 2sinh(pdg0001464pdg0004621) RHS diff is 2pdg0004621sin(pdg0001464)
3	0000111329: function is even number of inputs: 1; feeds: 3; outputs: 1 $#1$ is even with respect to $#2$, so replace $#1$ with $#3$ in Eq.~\ref{eq:#4}; yields Eq.~\ref{eq:#5}.	2394853829 $\exp(-i x) = \cos(-x)+i \sin(-x)$	0003413423 $\cos(-x)$ 0001030901 $\cos(x)$ 0004849392 $x$	4938429482 $\exp(-i x) = \cos(x)+i \sin(-x)$
4	0000111522: function is odd number of inputs: 1; feeds: 3; outputs: 1 $#1$ is odd with respect to $#2$, so replace $#1$ with $#3$ in Eq.~\ref{eq:#4}; yields Eq.~\ref{eq:#5}.	4938429482 $\exp(-i x) = \cos(x)+i \sin(-x)$	0002919191 $\sin(-x)$ 0003981813 $-\sin(x)$ 0003919391 $x$	4938429484 $\exp(-i x) = \cos(x)-i \sin(x)$
5	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}.	4938429484 $\exp(-i x) = \cos(x)-i \sin(x)$ 4938429483 $\exp(i x) = \cos(x)+i \sin(x)$		4742644828 $\exp(i x)+\exp(-i x) = 2 \cos(x)$	valid
6	0000111975: divide both sides by number of inputs: 1; feeds: 1; outputs: 1 Divide both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.	4742644828 $\exp(i x)+\exp(-i x) = 2 \cos(x)$	0004829194 $2$	3829492824 $\frac{1}{2}\left(\exp(i x)+\exp(-i x) \right) = \cos(x)$	valid
7	0000111268: swap LHS with RHS number of inputs: 1; feeds: 0; outputs: 1 Swap LHS of Eq.~\ref{eq:#1} with RHS; yields Eq.~\ref{eq:#2}.	3829492824 $\frac{1}{2}\left(\exp(i x)+\exp(-i x) \right) = \cos(x)$		4585932229 $\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)$	valid
8	0000111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\ref{eq:#1} is one of the final equations.	2103023049 $\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)$ 4585932229 $\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)$			no validation is available for declarations
9	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}.	4938429484 $\exp(-i x) = \cos(x)-i \sin(x)$	0003747849 $-1$	2123139121 $-\exp(-i x) = -\cos(x)+i \sin(x)$	valid
10	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}.	2123139121 $-\exp(-i x) = -\cos(x)+i \sin(x)$ 4938429483 $\exp(i x) = \cos(x)+i \sin(x)$		3942849294 $\exp(i x)-\exp(-i x) = 2 i \sin(x)$	valid
11	0000111975: divide both sides by number of inputs: 1; feeds: 1; outputs: 1 Divide both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.	3942849294 $\exp(i x)-\exp(-i x) = 2 i \sin(x)$	0001921933 $2 i$	4843995999 $\frac{1}{2 i}\left(\exp(i x)-\exp(-i x) \right) = \sin(x)$	valid
12	0000111268: swap LHS with RHS number of inputs: 1; feeds: 0; outputs: 1 Swap LHS of Eq.~\ref{eq:#1} with RHS; yields Eq.~\ref{eq:#2}.	4843995999 $\frac{1}{2 i}\left(\exp(i x)-\exp(-i x) \right) = \sin(x)$		2103023049 $\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)$	valid

Symbols used in Euler equation: trigonometric relations

0000004621: $ i $
0000001464: $ x $

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0.003 seconds for pdg_app/to_review_derivation be99eea8-2506-4419-b72f-f855e70b194f
0.004 seconds for compute/get_dict_of_steps_in_derivation: steps_in_this_derivation9bc8993f-900f-4766-ab64-3f4075348379
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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.			4938429483 \(\exp(i x) = \cos(x)+i \sin(x)\)	no validation is available for declarations
2	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}.	4938429483 \(\exp(i x) = \cos(x)+i \sin(x)\)	0003949052 \(-x\) 0002393922 \(x\)	2394853829 \(\exp(-i x) = \cos(-x)+i \sin(-x)\)	LHS diff is 2sinh(pdg0001464pdg0004621) RHS diff is 2pdg0004621sin(pdg0001464)
3	0000111329: function is even number of inputs: 1; feeds: 3; outputs: 1 $#1$ is even with respect to $#2$, so replace $#1$ with $#3$ in Eq.~\ref{eq:#4}; yields Eq.~\ref{eq:#5}.	2394853829 \(\exp(-i x) = \cos(-x)+i \sin(-x)\)	0003413423 \(\cos(-x)\) 0001030901 \(\cos(x)\) 0004849392 \(x\)	4938429482 \(\exp(-i x) = \cos(x)+i \sin(-x)\)
4	0000111522: function is odd number of inputs: 1; feeds: 3; outputs: 1 $#1$ is odd with respect to $#2$, so replace $#1$ with $#3$ in Eq.~\ref{eq:#4}; yields Eq.~\ref{eq:#5}.	4938429482 \(\exp(-i x) = \cos(x)+i \sin(-x)\)	0002919191 \(\sin(-x)\) 0003981813 \(-\sin(x)\) 0003919391 \(x\)	4938429484 \(\exp(-i x) = \cos(x)-i \sin(x)\)
5	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}.	4938429484 \(\exp(-i x) = \cos(x)-i \sin(x)\) 4938429483 \(\exp(i x) = \cos(x)+i \sin(x)\)		4742644828 \(\exp(i x)+\exp(-i x) = 2 \cos(x)\)	valid
6	0000111975: divide both sides by number of inputs: 1; feeds: 1; outputs: 1 Divide both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.	4742644828 \(\exp(i x)+\exp(-i x) = 2 \cos(x)\)	0004829194 \(2\)	3829492824 \(\frac{1}{2}\left(\exp(i x)+\exp(-i x) \right) = \cos(x)\)	valid
7	0000111268: swap LHS with RHS number of inputs: 1; feeds: 0; outputs: 1 Swap LHS of Eq.~\ref{eq:#1} with RHS; yields Eq.~\ref{eq:#2}.	3829492824 \(\frac{1}{2}\left(\exp(i x)+\exp(-i x) \right) = \cos(x)\)		4585932229 \(\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)\)	valid
8	0000111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\ref{eq:#1} is one of the final equations.	2103023049 \(\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)\) 4585932229 \(\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)\)			no validation is available for declarations
9	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}.	4938429484 \(\exp(-i x) = \cos(x)-i \sin(x)\)	0003747849 \(-1\)	2123139121 \(-\exp(-i x) = -\cos(x)+i \sin(x)\)	valid
10	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}.	2123139121 \(-\exp(-i x) = -\cos(x)+i \sin(x)\) 4938429483 \(\exp(i x) = \cos(x)+i \sin(x)\)		3942849294 \(\exp(i x)-\exp(-i x) = 2 i \sin(x)\)	valid
11	0000111975: divide both sides by number of inputs: 1; feeds: 1; outputs: 1 Divide both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.	3942849294 \(\exp(i x)-\exp(-i x) = 2 i \sin(x)\)	0001921933 \(2 i\)	4843995999 \(\frac{1}{2 i}\left(\exp(i x)-\exp(-i x) \right) = \sin(x)\)	valid
12	0000111268: swap LHS with RHS number of inputs: 1; feeds: 0; outputs: 1 Swap LHS of Eq.~\ref{eq:#1} with RHS; yields Eq.~\ref{eq:#2}.	4843995999 \(\frac{1}{2 i}\left(\exp(i x)-\exp(-i x) \right) = \sin(x)\)		2103023049 \(\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)\)	valid