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Review identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation

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.			2103023049 $\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)$	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}.	2103023049 $\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)$	9110536742 $2 x$ 4961662865 $x$	8483686863 $\sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x) \right)$	LHS diff is sin(pdg0001464) - sin(2pdg0001464) RHS diff is (-exp(4pdg0001464pdg0004621) + exp(3pdg0001464pdg0004621) - exp(pdg0001464pdg0004621) + 1)exp(-2pdg0001464pdg0004621)/(2pdg0004621)
3	0000111981: 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
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}.	4585932229 $\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)$ 2103023049 $\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)$		3470587782 $\sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)$	valid
5	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}.	3470587782 $\sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)$	8642992037 $2$	9894826550 $2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \left(\exp(i x)+\exp(-i x) \right)$	valid
6	0000111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	9894826550 $2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \left(\exp(i x)+\exp(-i x) \right)$		8699789241 $2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)$	valid
7	0000111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	8699789241 $2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)$		9180861128 $2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)$	valid
8	0000111863: RHS of expr 1 equals RHS of expr 2 number of inputs: 2; feeds: 0; outputs: 1 RHS of Eq.~\ref{eq:#1} is equal to RHS of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	8483686863 $\sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x) \right)$ 9180861128 $2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)$		2405307372 $\sin(2 x) = 2 \sin(x) \cos(x)$	valid
9	0000111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\ref{eq:#1} is one of the final equations.	2405307372 $\sin(2 x) = 2 \sin(x) \cos(x)$			no validation is available for declarations

Symbols used in identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation

0000004621: $ i $
0000001464: $ x $

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0.044 seconds for pdg_app/to_review_derivation af0c5e85-a87b-45cf-8750-99a010f616ef
0.021 seconds for compute/get_dict_of_steps_in_derivation: steps_in_this_derivatione0edaa66-c4ee-4ab5-b14f-c21e00a78862
0.026 seconds for compute/input_feed_output_infrule_for_step: get_inference_rule_connected_to_step_IDe0edaa66-c4ee-4ab5-b14f-c21e00a78862
0.014 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type HAS_INPUTe0edaa66-c4ee-4ab5-b14f-c21e00a78862
0.023 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type, HAS_FEEDe0edaa66-c4ee-4ab5-b14f-c21e00a78862
0.021 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type, HAS_OUTPUTe0edaa66-c4ee-4ab5-b14f-c21e00a78862
0.013 seconds for compute/get_dict_of_steps_in_derivation: get_sequence_index_for_stepe0edaa66-c4ee-4ab5-b14f-c21e00a78862
0.012 seconds for pdg_app/ af0c5e85-a87b-45cf-8750-99a010f616ef

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.			2103023049 \(\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)\)	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}.	2103023049 \(\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)\)	9110536742 \(2 x\) 4961662865 \(x\)	8483686863 \(\sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x) \right)\)	LHS diff is sin(pdg0001464) - sin(2pdg0001464) RHS diff is (-exp(4pdg0001464pdg0004621) + exp(3pdg0001464pdg0004621) - exp(pdg0001464pdg0004621) + 1)exp(-2pdg0001464pdg0004621)/(2pdg0004621)
3	0000111981: 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
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}.	4585932229 \(\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)\) 2103023049 \(\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)\)		3470587782 \(\sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)\)	valid
5	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}.	3470587782 \(\sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)\)	8642992037 \(2\)	9894826550 \(2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \left(\exp(i x)+\exp(-i x) \right)\)	valid
6	0000111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	9894826550 \(2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \left(\exp(i x)+\exp(-i x) \right)\)		8699789241 \(2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)\)	valid
7	0000111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	8699789241 \(2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)\)		9180861128 \(2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)\)	valid
8	0000111863: RHS of expr 1 equals RHS of expr 2 number of inputs: 2; feeds: 0; outputs: 1 RHS of Eq.~\ref{eq:#1} is equal to RHS of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	8483686863 \(\sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x) \right)\) 9180861128 \(2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)\)		2405307372 \(\sin(2 x) = 2 \sin(x) \cos(x)\)	valid
9	0000111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\ref{eq:#1} is one of the final equations.	2405307372 \(\sin(2 x) = 2 \sin(x) \cos(x)\)			no validation is available for declarations