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

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. 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}.
  1. 2103023049
    \(\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)\)
  1. 9110536742
    \(2 x\)
  2. 4961662865
    \(x\)
  1. 8483686863
    \(\sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x) \right)\)
 LHS diff is sin(pdg0001464) - sin(2*pdg0001464) RHS diff is (-exp(4*pdg0001464*pdg0004621) + exp(3*pdg0001464*pdg0004621) - exp(pdg0001464*pdg0004621) + 1)*exp(-2*pdg0001464*pdg0004621)/(2*pdg0004621)
3
  • 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
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. 4585932229
    \(\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)\)
  2. 2103023049
    \(\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)\)
  1. 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}.
  1. 3470587782
    \(\sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)\)
  1. 8642992037
    \(2\)
  1. 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}.
  1. 9894826550
    \(2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \left(\exp(i x)+\exp(-i x) \right)\)
  1. 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}.
  1. 8699789241
    \(2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)\)
  1. 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}.
  1. 8483686863
    \(\sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x) \right)\)
  2. 9180861128
    \(2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)\)
  1. 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.
  1. 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

Steps and expressions for identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation

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