## 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

$$x$$
$$i$$