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Review Euler equation: trig square root

abstract:
reference:

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 $\exp(i x) = \cos(x)+i \sin(x)$	0004582412 $x$ 0004089571 $2 x$	4838429483 $\exp(2 i x) = \cos(2 x)+i \sin(2 x)$	valid
3	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}.	4838429483 $\exp(2 i x) = \cos(2 x)+i \sin(2 x)$ 4598294821 $\exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2$		9483928192 $\cos(2 x) + i\sin(2 x) = (\cos(x))^2 + 2 i \cos(x) \sin(x) - (\sin(x))^2$	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}.	4938429483 $\exp(i x) = \cos(x)+i \sin(x)$ 4938429483 $\exp(i x) = \cos(x)+i \sin(x)$		4638429483 $\exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))$	valid
5	0000111546: expand RHS number of inputs: 1; feeds: 0; outputs: 1 Expand the RHS of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	4638429483 $\exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))$		4598294821 $\exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2$	LHS diff is 0 RHS diff is (pdg0004621*2 + 1)sin(pdg0001464)**2
6	0000111198: select real parts number of inputs: 1; feeds: 0; outputs: 1 Select real parts of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	9483928192 $\cos(2 x) + i\sin(2 x) = (\cos(x))^2 + 2 i \cos(x) \sin(x) - (\sin(x))^2$		9482928242 $\cos(2 x) = (\cos(x))^2 - (\sin(x))^2$	LHS diff is -cos(2pdg0001464) + cos(2re(pdg0001464))cosh(2im(pdg0001464)) + re(pdg0004621sin(2pdg0001464)) RHS diff is -cos(2pdg0001464) + 2cos(2re(pdg0001464))sinh(im(pdg0001464))*2 + cos(2re(pdg0001464)) + re(pdg0004621sin(2pdg0001464))
7	0000111530: add X to both sides number of inputs: 1; feeds: 1; outputs: 1 Add $#1$ to both sides of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	9482928242 $\cos(2 x) = (\cos(x))^2 - (\sin(x))^2$	0007894942 $(\sin(x))^2$	9482928243 $\cos(2 x) + (\sin(x))^2 = (\cos(x))^2$	valid
8	0000111299: declare identity number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an identity.			5832984291 $(\sin(x))^2 + (\cos(x))^2 = 1$	no validation is available for declarations
9	0000111282: subtract X from both sides number of inputs: 1; feeds: 1; outputs: 1 Subtract $#1$ from both sides of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	5832984291 $(\sin(x))^2 + (\cos(x))^2 = 1$	0001111111 $(\sin(x))^2$	3285732911 $(\cos(x))^2 = 1-(\sin(x))^2$	valid
10	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}.	9482928243 $\cos(2 x) + (\sin(x))^2 = (\cos(x))^2$		9482438243 $(\cos(x))^2 = \cos(2 x) + (\sin(x))^2$	valid
11	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}.	3285732911 $(\cos(x))^2 = 1-(\sin(x))^2$ 9482438243 $(\cos(x))^2 = \cos(2 x) + (\sin(x))^2$		4827492911 $\cos(2 x)+(\sin(x))^2 = 1 - (\sin(x))^2$	valid
12	0000111282: subtract X from both sides number of inputs: 1; feeds: 1; outputs: 1 Subtract $#1$ from both sides of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	4827492911 $\cos(2 x)+(\sin(x))^2 = 1 - (\sin(x))^2$	0006466214 $(\sin(x))^2$	1248277773 $\cos(2 x) = 1 - 2 (\sin(x))^2$	valid
13	0000111530: add X to both sides number of inputs: 1; feeds: 1; outputs: 1 Add $#1$ to both sides of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	1248277773 $\cos(2 x) = 1 - 2 (\sin(x))^2$	0007471778 $2(\sin(x))^2$	7572664728 $\cos(2 x) + 2 (\sin(x))^2 = 1$	valid
14	0000111282: subtract X from both sides number of inputs: 1; feeds: 1; outputs: 1 Subtract $#1$ from both sides of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	7572664728 $\cos(2 x) + 2 (\sin(x))^2 = 1$	0008842811 $\cos(2 x)$	9889984281 $2 (\sin(x))^2 = 1 - \cos(2 x)$	valid
15	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}.	9889984281 $2 (\sin(x))^2 = 1 - \cos(2 x)$	0003838111 $2$	9988949211 $(\sin(x))^2 = \frac{1 - \cos(2 x)}{2}$	valid
16	0000111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\ref{eq:#1} is one of the final equations.	9988949211 $(\sin(x))^2 = \frac{1 - \cos(2 x)}{2}$			no validation is available for declarations

Symbols used in Euler equation: trig square root

0000004621: $ i $
0000001464: $ x $
0000004037: $ x $

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0.004 seconds for pdg_app/to_review_derivation 4eef9579-d2b6-424f-84b2-3fa396d4d8f3
0.005 seconds for compute/get_dict_of_steps_in_derivation: steps_in_this_derivation3ce7707e-7b93-416f-a51b-f521218db2de
0.003 seconds for compute/input_feed_output_infrule_for_step: get_inference_rule_connected_to_step_ID3ce7707e-7b93-416f-a51b-f521218db2de
0.006 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type HAS_INPUT3ce7707e-7b93-416f-a51b-f521218db2de
0.003 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type, HAS_FEED3ce7707e-7b93-416f-a51b-f521218db2de
0.003 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type, HAS_OUTPUT3ce7707e-7b93-416f-a51b-f521218db2de
0.004 seconds for compute/get_dict_of_steps_in_derivation: get_sequence_index_for_step3ce7707e-7b93-416f-a51b-f521218db2de
0.005 seconds for pdg_app/ 4eef9579-d2b6-424f-84b2-3fa396d4d8f3

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 \(\exp(i x) = \cos(x)+i \sin(x)\)	0004582412 \(x\) 0004089571 \(2 x\)	4838429483 \(\exp(2 i x) = \cos(2 x)+i \sin(2 x)\)	valid
3	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}.	4838429483 \(\exp(2 i x) = \cos(2 x)+i \sin(2 x)\) 4598294821 \(\exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2\)		9483928192 \(\cos(2 x) + i\sin(2 x) = (\cos(x))^2 + 2 i \cos(x) \sin(x) - (\sin(x))^2\)	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}.	4938429483 \(\exp(i x) = \cos(x)+i \sin(x)\) 4938429483 \(\exp(i x) = \cos(x)+i \sin(x)\)		4638429483 \(\exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))\)	valid
5	0000111546: expand RHS number of inputs: 1; feeds: 0; outputs: 1 Expand the RHS of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	4638429483 \(\exp(2 i x) = (\cos(x)+ i \sin(x))(\cos(x)+ i \sin(x))\)		4598294821 \(\exp(2 i x) = (\cos(x))^2+2i\cos(x)\sin(x)-(\sin(x))^2\)	LHS diff is 0 RHS diff is (pdg0004621*2 + 1)sin(pdg0001464)**2
6	0000111198: select real parts number of inputs: 1; feeds: 0; outputs: 1 Select real parts of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	9483928192 \(\cos(2 x) + i\sin(2 x) = (\cos(x))^2 + 2 i \cos(x) \sin(x) - (\sin(x))^2\)		9482928242 \(\cos(2 x) = (\cos(x))^2 - (\sin(x))^2\)	LHS diff is -cos(2pdg0001464) + cos(2re(pdg0001464))cosh(2im(pdg0001464)) + re(pdg0004621sin(2pdg0001464)) RHS diff is -cos(2pdg0001464) + 2cos(2re(pdg0001464))sinh(im(pdg0001464))*2 + cos(2re(pdg0001464)) + re(pdg0004621sin(2pdg0001464))
7	0000111530: add X to both sides number of inputs: 1; feeds: 1; outputs: 1 Add $#1$ to both sides of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	9482928242 \(\cos(2 x) = (\cos(x))^2 - (\sin(x))^2\)	0007894942 \((\sin(x))^2\)	9482928243 \(\cos(2 x) + (\sin(x))^2 = (\cos(x))^2\)	valid
8	0000111299: declare identity number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an identity.			5832984291 \((\sin(x))^2 + (\cos(x))^2 = 1\)	no validation is available for declarations
9	0000111282: subtract X from both sides number of inputs: 1; feeds: 1; outputs: 1 Subtract $#1$ from both sides of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	5832984291 \((\sin(x))^2 + (\cos(x))^2 = 1\)	0001111111 \((\sin(x))^2\)	3285732911 \((\cos(x))^2 = 1-(\sin(x))^2\)	valid
10	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}.	9482928243 \(\cos(2 x) + (\sin(x))^2 = (\cos(x))^2\)		9482438243 \((\cos(x))^2 = \cos(2 x) + (\sin(x))^2\)	valid
11	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}.	3285732911 \((\cos(x))^2 = 1-(\sin(x))^2\) 9482438243 \((\cos(x))^2 = \cos(2 x) + (\sin(x))^2\)		4827492911 \(\cos(2 x)+(\sin(x))^2 = 1 - (\sin(x))^2\)	valid
12	0000111282: subtract X from both sides number of inputs: 1; feeds: 1; outputs: 1 Subtract $#1$ from both sides of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	4827492911 \(\cos(2 x)+(\sin(x))^2 = 1 - (\sin(x))^2\)	0006466214 \((\sin(x))^2\)	1248277773 \(\cos(2 x) = 1 - 2 (\sin(x))^2\)	valid
13	0000111530: add X to both sides number of inputs: 1; feeds: 1; outputs: 1 Add $#1$ to both sides of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	1248277773 \(\cos(2 x) = 1 - 2 (\sin(x))^2\)	0007471778 \(2(\sin(x))^2\)	7572664728 \(\cos(2 x) + 2 (\sin(x))^2 = 1\)	valid
14	0000111282: subtract X from both sides number of inputs: 1; feeds: 1; outputs: 1 Subtract $#1$ from both sides of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	7572664728 \(\cos(2 x) + 2 (\sin(x))^2 = 1\)	0008842811 \(\cos(2 x)\)	9889984281 \(2 (\sin(x))^2 = 1 - \cos(2 x)\)	valid
15	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}.	9889984281 \(2 (\sin(x))^2 = 1 - \cos(2 x)\)	0003838111 \(2\)	9988949211 \((\sin(x))^2 = \frac{1 - \cos(2 x)}{2}\)	valid
16	0000111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\ref{eq:#1} is one of the final equations.	9988949211 \((\sin(x))^2 = \frac{1 - \cos(2 x)}{2}\)			no validation is available for declarations