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review derivation: identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
4 multiply expr 1 by expr 2
  1. 2103023049; locally 6060683:
    \(\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)\)
    \(\sin{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}}}{2 pdg_{4621}}\)
  2. 4585932229; locally 5011637:
    \(\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)\)
    \(\cos{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464} pdg_{4621}}}{2} + \frac{e^{- pdg_{1464} pdg_{4621}}}{2}\)
  1. 3470587782; locally 6350246:
    \(\sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)\)
    \(\sin{\left(pdg_{1464} \right)} \cos{\left(pdg_{1464} \right)} = \frac{\left(\frac{e^{pdg_{1464} pdg_{4621}}}{2} + \frac{e^{- pdg_{1464} pdg_{4621}}}{2}\right) \left(e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}}\right)}{2 pdg_{4621}}\)
valid 2103023049:
4585932229:
3470587782:
2103023049:
4585932229:
3470587782:
8 RHS of expr 1 equals RHS of expr 2
  1. 9180861128; locally 6229292:
    \(2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)\)
    \(2 \sin{\left(pdg_{1464} \right)} \cos{\left(pdg_{1464} \right)} = \frac{e^{2 pdg_{1464} pdg_{4621}} - e^{- 2 pdg_{1464} pdg_{4621}}}{2 pdg_{4621}}\)
  2. 8483686863; locally 1414263:
    \(\sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x) \right)\)
    \(\sin{\left(2 pdg_{1464} \right)} = \frac{e^{2 pdg_{1464} pdg_{4621}} - e^{- 2 pdg_{1464} pdg_{4621}}}{2 pdg_{4621}}\)
  1. 2405307372; locally 7647794:
    \(\sin(2 x) = 2 \sin(x) \cos(x)\)
    \(\sin{\left(2 pdg_{1464} \right)} = 2 \sin{\left(pdg_{1464} \right)} \cos{\left(pdg_{1464} \right)}\)
valid 9180861128:
8483686863:
2405307372:
9180861128:
8483686863:
2405307372:
1 declare initial expr
  1. 2103023049; locally 6060683:
    \(\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)\)
    \(\sin{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}}}{2 pdg_{4621}}\)
no validation is available for declarations 2103023049:
2103023049:
3 declare initial expr
  1. 4585932229; locally 5011637:
    \(\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)\)
    \(\cos{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464} pdg_{4621}}}{2} + \frac{e^{- pdg_{1464} pdg_{4621}}}{2}\)
no validation is available for declarations 4585932229:
4585932229:
2 change variable X to Y
  1. 2103023049; locally 6060683:
    \(\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)\)
    \(\sin{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}}}{2 pdg_{4621}}\)
  1. 4961662865:
    \(x\)
    \(pdg_{1464}\)
  2. 9110536742:
    \(2 x\)
    \(2 pdg_{1464}\)
  1. 8483686863; locally 1414263:
    \(\sin(2 x) = \frac{1}{2i}\left(\exp(i 2 x)-\exp(-i 2 x) \right)\)
    \(\sin{\left(2 pdg_{1464} \right)} = \frac{e^{2 pdg_{1464} pdg_{4621}} - e^{- 2 pdg_{1464} pdg_{4621}}}{2 pdg_{4621}}\)
valid 2103023049:
8483686863:
2103023049:
8483686863:
7 simplify
  1. 8699789241; locally 5714636:
    \(2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)\)
    \(2 \sin{\left(pdg_{1464} \right)} \cos{\left(pdg_{1464} \right)} = \frac{e^{2 pdg_{1464} pdg_{4621}} - e^{- 2 pdg_{1464} pdg_{4621}}}{2 pdg_{4621}}\)
  1. 9180861128; locally 6229292:
    \(2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - \exp(-i 2 x) \right)\)
    \(2 \sin{\left(pdg_{1464} \right)} \cos{\left(pdg_{1464} \right)} = \frac{e^{2 pdg_{1464} pdg_{4621}} - e^{- 2 pdg_{1464} pdg_{4621}}}{2 pdg_{4621}}\)
valid 8699789241: error for dim with 8699789241
9180861128:
8699789241: N/A
9180861128:
9 declare final expr
  1. 2405307372; locally 7647794:
    \(\sin(2 x) = 2 \sin(x) \cos(x)\)
    \(\sin{\left(2 pdg_{1464} \right)} = 2 \sin{\left(pdg_{1464} \right)} \cos{\left(pdg_{1464} \right)}\)
no validation is available for declarations 2405307372:
2405307372:
5 multiply both sides by
  1. 3470587782; locally 6350246:
    \(\sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)\)
    \(\sin{\left(pdg_{1464} \right)} \cos{\left(pdg_{1464} \right)} = \frac{\left(\frac{e^{pdg_{1464} pdg_{4621}}}{2} + \frac{e^{- pdg_{1464} pdg_{4621}}}{2}\right) \left(e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}}\right)}{2 pdg_{4621}}\)
  1. 8642992037:
    \(2\)
    \(2\)
  1. 9894826550; locally 7867574:
    \(2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \left(\exp(i x)+\exp(-i x) \right)\)
    \(2 \sin{\left(pdg_{1464} \right)} \cos{\left(pdg_{1464} \right)} = \frac{\left(e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}}\right) \left(e^{pdg_{1464} pdg_{4621}} + e^{- pdg_{1464} pdg_{4621}}\right)}{2 pdg_{4621}}\)
valid 3470587782:
9894826550: error for dim with 9894826550
3470587782:
9894826550: N/A
6 simplify
  1. 9894826550; locally 7867574:
    \(2 \sin(x) \cos(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right) \left(\exp(i x)+\exp(-i x) \right)\)
    \(2 \sin{\left(pdg_{1464} \right)} \cos{\left(pdg_{1464} \right)} = \frac{\left(e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}}\right) \left(e^{pdg_{1464} pdg_{4621}} + e^{- pdg_{1464} pdg_{4621}}\right)}{2 pdg_{4621}}\)
  1. 8699789241; locally 5714636:
    \(2 \sin(x) \cos(x) = \frac{1}{2 i} \left( \exp(i 2 x) - 1 + 1 - \exp(-i 2 x) \right)\)
    \(2 \sin{\left(pdg_{1464} \right)} \cos{\left(pdg_{1464} \right)} = \frac{e^{2 pdg_{1464} pdg_{4621}} - e^{- 2 pdg_{1464} pdg_{4621}}}{2 pdg_{4621}}\)
valid 9894826550: error for dim with 9894826550
8699789241: error for dim with 8699789241
9894826550: N/A
8699789241: N/A
Physics Derivation Graph: Steps for identity sin(2 x) = 2 sin(x) cos(x) using Euler's equation

Symbols for this derivation

See also all 215 symbols
symbol ID category latex scope dimension name value Used in derivations references
1464 variable x
\(x\)
['real'] dimensionless 140
4621 variable i
\(i\)
['imaginary'] dimensionless imaginary unit 74
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