## review derivation: hyperbolic trigonometric identities

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Notes for this derivation:
http://www.physics.miami.edu/~nearing/mathmethods/mathematical_methods-one.pdf

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
11 declare initial expr
1. 2103023049; locally 3077940:
$$\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)$$
$$\sin{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}}}{2 pdg_{4621}}$$
no validation is available for declarations 2103023049:
2103023049:
26 simplify
1. 1128605625; locally 6426652:
$${\rm sech}^2\ x + \tanh^2(x) = \frac{4}{\left(\exp(x)+\exp(-x)\right)^2} + \frac{\left(\exp(x)-\exp(-x)\right)^2}{\left(\exp(x)+\exp(-x)\right)^2}$$
$$\tanh^{2}{\left(pdg_{1464} \right)} + \operatorname{sech}^{2}{\left(pdg_{1464} \right)} = \frac{\left(e^{pdg_{1464}} - e^{- pdg_{1464}}\right)^{2}}{\left(e^{pdg_{1464}} + e^{- pdg_{1464}}\right)^{2}} + \frac{4}{\left(e^{pdg_{1464}} + e^{- pdg_{1464}}\right)^{2}}$$
1. 4830221561; locally 6070484:
$${\rm sech}^2\ x + \tanh^2(x) = \frac{4+\left(\exp(2x)-1-1+\exp(-2x)\right)}{\left(\exp(x)+\exp(-x)\right)^2}$$
$$\tanh^{2}{\left(pdg_{1464} \right)} + \operatorname{sech}^{2}{\left(pdg_{1464} \right)} = \frac{e^{2 pdg_{1464}} + 2 + e^{- 2 pdg_{1464}}}{\left(e^{pdg_{1464}} + e^{- pdg_{1464}}\right)^{2}}$$
valid 1128605625:
4830221561:
1128605625:
4830221561:
25 add expr 1 to expr 2
1. 3868998312; locally 5395954:
$${\rm sech}^2\ x = \frac{4}{\left(\exp(x)+\exp(-x)\right)^2}$$
$$\operatorname{sech}^{2}{\left(pdg_{1464} \right)} = \frac{4}{\left(e^{pdg_{1464}} + e^{- pdg_{1464}}\right)^{2}}$$
2. 2121790783; locally 9317216:
$$\tanh^2(x) = \frac{ \left(\exp(x)-\exp(-x)\right)^2}{\left(\exp(x)+\exp(-x)\right)^2}$$
$$\tanh^{2}{\left(pdg_{1464} \right)} = \frac{\left(e^{pdg_{1464}} - e^{- pdg_{1464}}\right)^{2}}{\left(e^{pdg_{1464}} + e^{- pdg_{1464}}\right)^{2}}$$
1. 1128605625; locally 6426652:
$${\rm sech}^2\ x + \tanh^2(x) = \frac{4}{\left(\exp(x)+\exp(-x)\right)^2} + \frac{\left(\exp(x)-\exp(-x)\right)^2}{\left(\exp(x)+\exp(-x)\right)^2}$$
$$\tanh^{2}{\left(pdg_{1464} \right)} + \operatorname{sech}^{2}{\left(pdg_{1464} \right)} = \frac{\left(e^{pdg_{1464}} - e^{- pdg_{1464}}\right)^{2}}{\left(e^{pdg_{1464}} + e^{- pdg_{1464}}\right)^{2}} + \frac{4}{\left(e^{pdg_{1464}} + e^{- pdg_{1464}}\right)^{2}}$$
valid 3868998312:
2121790783:
1128605625:
3868998312:
2121790783:
1128605625:
15 declare initial expr
1. 4585932229; locally 4731536:
$$\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)$$
$$\cos{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464} pdg_{4621}}}{2} + \frac{e^{- pdg_{1464} pdg_{4621}}}{2}$$
no validation is available for declarations 4585932229:
4585932229:
19 substitute LHS of expr 1 into expr 2
1. 6404535647; locally 4319733:
$$\cosh x = \frac{\exp(x) + \exp(-x)}{2}$$
$$\cosh{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464}}}{2} + \frac{e^{- pdg_{1464}}}{2}$$
2. 7731226616; locally 3909583:
$${\rm sech}\ x = \frac{1}{\cosh x}$$
$$\operatorname{sech}{\left(pdg_{1464} \right)} = \frac{1}{\cosh{\left(pdg_{1464} \right)}}$$
1. 4166155526; locally 7222556:
$${\rm sech}\ x = \frac{2}{\exp(x)+\exp(-x)}$$
$$\operatorname{sech}{\left(pdg_{1464} \right)} = \frac{2}{e^{pdg_{1464}} + e^{- pdg_{1464}}}$$
valid 6404535647: error for dim with 6404535647
7731226616:
4166155526:
6404535647: N/A
7731226616:
4166155526:
21 substitute LHS of expr 1 into expr 2
1. 1038566242; locally 3145608:
$$\sinh x = \frac{\exp(x) - \exp(-x)}{2}$$
$$\sinh{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464}}}{2} - \frac{e^{- pdg_{1464}}}{2}$$
2. 4872163189; locally 3867418:
$$\tanh(x) = \frac{\sinh(x)}{\cosh(x)}$$
$$\tanh{\left(pdg_{1464} \right)} = \frac{\sinh{\left(pdg_{1464} \right)}}{\cosh{\left(pdg_{1464} \right)}}$$
1. 2902772962; locally 6831354:
$$\tanh(x) = \frac{\frac{1}{2}\left( \exp(x)-\exp(-x) \right)}{\cosh(x)}$$
$$\tanh{\left(pdg_{1464} \right)} = \frac{\frac{e^{pdg_{1464}}}{2} - \frac{e^{- pdg_{1464}}}{2}}{\cosh{\left(pdg_{1464} \right)}}$$
valid 1038566242:
4872163189:
2902772962:
1038566242:
4872163189:
2902772962:
27 simplify
1. 4830221561; locally 6070484:
$${\rm sech}^2\ x + \tanh^2(x) = \frac{4+\left(\exp(2x)-1-1+\exp(-2x)\right)}{\left(\exp(x)+\exp(-x)\right)^2}$$
$$\tanh^{2}{\left(pdg_{1464} \right)} + \operatorname{sech}^{2}{\left(pdg_{1464} \right)} = \frac{e^{2 pdg_{1464}} + 2 + e^{- 2 pdg_{1464}}}{\left(e^{pdg_{1464}} + e^{- pdg_{1464}}\right)^{2}}$$
1. 5866629429; locally 8702257:
$${\rm sech}^2\ x + \tanh^2(x) = 1$$
$$\tanh^{2}{\left(pdg_{1464} \right)} + \operatorname{sech}^{2}{\left(pdg_{1464} \right)} = 1$$
valid 4830221561:
5866629429:
4830221561:
5866629429:
2 declare initial expr
1. 1038566242; locally 3145608:
$$\sinh x = \frac{\exp(x) - \exp(-x)}{2}$$
$$\sinh{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464}}}{2} - \frac{e^{- pdg_{1464}}}{2}$$
no validation is available for declarations 1038566242:
1038566242:
12 change variable X to Y
1. 2103023049; locally 3077940:
$$\sin(x) = \frac{1}{2i}\left(\exp(i x)-\exp(-i x) \right)$$
$$\sin{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464} pdg_{4621}} - e^{- pdg_{1464} pdg_{4621}}}{2 pdg_{4621}}$$
1. 6976493023:
$$x$$
$$pdg_{1464}$$
2. 7159989263:
$$i x$$
$$pdg_{1464} pdg_{4621}$$
1. 4878728014; locally 5823930:
$$\sin(i x) = \frac{1}{2i}\left(\exp(-x) - \exp(x) \right)$$
$$\sin{\left(pdg_{1464} pdg_{4621} \right)} = \frac{- e^{pdg_{1464}} + e^{- pdg_{1464}}}{2 pdg_{4621}}$$
LHS diff is 0 RHS diff is exp(pdg1464)/(2*pdg4621) + exp(pdg1464*pdg4621**2)/(2*pdg4621) - exp(-pdg1464*pdg4621**2)/(2*pdg4621) - exp(-pdg1464)/(2*pdg4621) 2103023049:
4878728014:
2103023049:
4878728014:
24 multiply expr 1 by expr 2
1. 4166155526; locally 7222556:
$${\rm sech}\ x = \frac{2}{\exp(x)+\exp(-x)}$$
$$\operatorname{sech}{\left(pdg_{1464} \right)} = \frac{2}{e^{pdg_{1464}} + e^{- pdg_{1464}}}$$
2. 4166155526; locally 7222556:
$${\rm sech}\ x = \frac{2}{\exp(x)+\exp(-x)}$$
$$\operatorname{sech}{\left(pdg_{1464} \right)} = \frac{2}{e^{pdg_{1464}} + e^{- pdg_{1464}}}$$
1. 3868998312; locally 5395954:
$${\rm sech}^2\ x = \frac{4}{\left(\exp(x)+\exp(-x)\right)^2}$$
$$\operatorname{sech}^{2}{\left(pdg_{1464} \right)} = \frac{4}{\left(e^{pdg_{1464}} + e^{- pdg_{1464}}\right)^{2}}$$
valid 4166155526:
4166155526:
3868998312:
4166155526:
4166155526:
3868998312:
28 declare final expr
1. 5866629429; locally 8702257:
$${\rm sech}^2\ x + \tanh^2(x) = 1$$
$$\tanh^{2}{\left(pdg_{1464} \right)} + \operatorname{sech}^{2}{\left(pdg_{1464} \right)} = 1$$
no validation is available for declarations 5866629429:
5866629429:
14 multiply both sides by
1. 5323719091; locally 2016533:
$$i \sinh x = \frac{1}{2i} \left( \exp(-x) - \exp(x) \right)$$
$$pdg_{4621} \sinh{\left(pdg_{1464} \right)} = \frac{- e^{pdg_{1464}} + e^{- pdg_{1464}}}{2 pdg_{4621}}$$
1. 9885190237:
$$i$$
$$pdg_{4621}$$
1. 1038566242; locally 3145608:
$$\sinh x = \frac{\exp(x) - \exp(-x)}{2}$$
$$\sinh{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464}}}{2} - \frac{e^{- pdg_{1464}}}{2}$$
LHS diff is (pdg4621**2 - 1)*sinh(pdg1464) RHS diff is -2*sinh(pdg1464) 5323719091:
1038566242:
5323719091:
1038566242:
7 simplify
1. 2762326680; locally 4009221:
$$\cosh^2 x - \sinh^2 x = \frac{1}{4}\left( \exp(2x)+1+1+\exp(-2x) - \left(\exp(2x)-1-1+\exp(-2x)\right) \right)$$
$$- \sinh^{2}{\left(pdg_{1464} \right)} + \cosh^{2}{\left(pdg_{1464} \right)} = 1$$
1. 9413609246; locally 6300507:
$$\cosh^2 x - \sinh^2 x = 1$$
$$- \sinh^{2}{\left(pdg_{1464} \right)} + \cosh^{2}{\left(pdg_{1464} \right)} = 1$$
valid 2762326680:
9413609246:
2762326680:
9413609246:
3 multiply expr 1 by expr 2
1. 1038566242; locally 3145608:
$$\sinh x = \frac{\exp(x) - \exp(-x)}{2}$$
$$\sinh{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464}}}{2} - \frac{e^{- pdg_{1464}}}{2}$$
2. 1038566242; locally 3145608:
$$\sinh x = \frac{\exp(x) - \exp(-x)}{2}$$
$$\sinh{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464}}}{2} - \frac{e^{- pdg_{1464}}}{2}$$
1. 6031385191; locally 7844176:
$$\sinh^2 x = \left(\frac{\exp(x) - \exp(-x)}{2}\right)\left(\frac{\exp(x) - \exp(-x)}{2}\right)$$
$$\sinh^{2}{\left(pdg_{1464} \right)} = \left(\frac{e^{pdg_{1464}}}{2} - \frac{e^{- pdg_{1464}}}{2}\right)^{2}$$
valid 1038566242:
1038566242:
6031385191:
1038566242:
1038566242:
6031385191:
16 change variable X to Y
1. 4585932229; locally 4731536:
$$\cos(x) = \frac{1}{2}\left(\exp(i x)+\exp(-i x) \right)$$
$$\cos{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464} pdg_{4621}}}{2} + \frac{e^{- pdg_{1464} pdg_{4621}}}{2}$$
1. 7453225570:
$$x$$
$$pdg_{1464}$$
2. 1716984328:
$$i x$$
$$pdg_{1464} pdg_{4621}$$
1. 8651044341; locally 6479977:
$$\cos(i x) = \frac{1}{2} \left( \exp(-x) + \exp(x) \right)$$
$$\cos{\left(pdg_{1464} pdg_{4621} \right)} = \frac{e^{pdg_{1464}}}{2} + \frac{e^{- pdg_{1464}}}{2}$$
LHS diff is 0 RHS diff is -cosh(pdg1464) + cosh(pdg1464*pdg4621**2) 4585932229:
8651044341:
4585932229:
8651044341:
8 declare final expr
1. 9413609246; locally 6300507:
$$\cosh^2 x - \sinh^2 x = 1$$
$$- \sinh^{2}{\left(pdg_{1464} \right)} + \cosh^{2}{\left(pdg_{1464} \right)} = 1$$
no validation is available for declarations 9413609246:
9413609246:
10 declare initial expr
1. 8418527415; locally 5377003:
$$\sin(i x) = i \sinh(x)$$
$$\sin{\left(pdg_{1464} pdg_{4621} \right)} = pdg_{4621} \sinh{\left(pdg_{1464} \right)}$$
no validation is available for declarations 8418527415:
8418527415:
22 substitute LHS of expr 1 into expr 2
1. 6404535647; locally 4319733:
$$\cosh x = \frac{\exp(x) + \exp(-x)}{2}$$
$$\cosh{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464}}}{2} + \frac{e^{- pdg_{1464}}}{2}$$
2. 2902772962; locally 6831354:
$$\tanh(x) = \frac{\frac{1}{2}\left( \exp(x)-\exp(-x) \right)}{\cosh(x)}$$
$$\tanh{\left(pdg_{1464} \right)} = \frac{\frac{e^{pdg_{1464}}}{2} - \frac{e^{- pdg_{1464}}}{2}}{\cosh{\left(pdg_{1464} \right)}}$$
1. 5349669879; locally 5313211:
$$\tanh(x) = \frac{ \exp(x)-\exp(-x)}{\exp(x)+\exp(-x)}$$
$$\tanh{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464}} - e^{- pdg_{1464}}}{e^{pdg_{1464}} + e^{- pdg_{1464}}}$$
valid 6404535647: error for dim with 6404535647
2902772962:
5349669879:
6404535647: N/A
2902772962:
5349669879:
13 LHS of expr 1 equals LHS of expr 2
1. 4878728014; locally 5823930:
$$\sin(i x) = \frac{1}{2i}\left(\exp(-x) - \exp(x) \right)$$
$$\sin{\left(pdg_{1464} pdg_{4621} \right)} = \frac{- e^{pdg_{1464}} + e^{- pdg_{1464}}}{2 pdg_{4621}}$$
2. 8418527415; locally 5377003:
$$\sin(i x) = i \sinh(x)$$
$$\sin{\left(pdg_{1464} pdg_{4621} \right)} = pdg_{4621} \sinh{\left(pdg_{1464} \right)}$$
1. 5323719091; locally 2016533:
$$i \sinh x = \frac{1}{2i} \left( \exp(-x) - \exp(x) \right)$$
$$pdg_{4621} \sinh{\left(pdg_{1464} \right)} = \frac{- e^{pdg_{1464}} + e^{- pdg_{1464}}}{2 pdg_{4621}}$$
input diff is 0 diff is pdg4621*sinh(pdg1464) + exp(pdg1464)/(2*pdg4621) - exp(-pdg1464)/(2*pdg4621) diff is -pdg4621*sinh(pdg1464) - exp(pdg1464)/(2*pdg4621) + exp(-pdg1464)/(2*pdg4621) 4878728014:
8418527415:
5323719091:
4878728014:
8418527415:
5323719091:
20 declare initial expr
1. 4872163189; locally 3867418:
$$\tanh(x) = \frac{\sinh(x)}{\cosh(x)}$$
$$\tanh{\left(pdg_{1464} \right)} = \frac{\sinh{\left(pdg_{1464} \right)}}{\cosh{\left(pdg_{1464} \right)}}$$
no validation is available for declarations 4872163189:
4872163189:
17 LHS of expr 1 equals LHS of expr 2
1. 8747785338; locally 7404421:
$$\cos(i x) = \cosh(x)$$
$$\cos{\left(pdg_{1464} pdg_{4621} \right)} = \cosh{\left(pdg_{1464} \right)}$$
2. 8651044341; locally 6479977:
$$\cos(i x) = \frac{1}{2} \left( \exp(-x) + \exp(x) \right)$$
$$\cos{\left(pdg_{1464} pdg_{4621} \right)} = \frac{e^{pdg_{1464}}}{2} + \frac{e^{- pdg_{1464}}}{2}$$
1. 6404535647; locally 4319733:
$$\cosh x = \frac{\exp(x) + \exp(-x)}{2}$$
$$\cosh{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464}}}{2} + \frac{e^{- pdg_{1464}}}{2}$$
valid 8747785338:
8651044341:
6404535647: error for dim with 6404535647
8747785338:
8651044341:
6404535647: N/A
18 declare initial expr
1. 7731226616; locally 3909583:
$${\rm sech}\ x = \frac{1}{\cosh x}$$
$$\operatorname{sech}{\left(pdg_{1464} \right)} = \frac{1}{\cosh{\left(pdg_{1464} \right)}}$$
no validation is available for declarations 7731226616:
7731226616:
23 multiply expr 1 by expr 2
1. 5349669879; locally 5313211:
$$\tanh(x) = \frac{ \exp(x)-\exp(-x)}{\exp(x)+\exp(-x)}$$
$$\tanh{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464}} - e^{- pdg_{1464}}}{e^{pdg_{1464}} + e^{- pdg_{1464}}}$$
2. 5349669879; locally 5313211:
$$\tanh(x) = \frac{ \exp(x)-\exp(-x)}{\exp(x)+\exp(-x)}$$
$$\tanh{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464}} - e^{- pdg_{1464}}}{e^{pdg_{1464}} + e^{- pdg_{1464}}}$$
1. 2121790783; locally 9317216:
$$\tanh^2(x) = \frac{ \left(\exp(x)-\exp(-x)\right)^2}{\left(\exp(x)+\exp(-x)\right)^2}$$
$$\tanh^{2}{\left(pdg_{1464} \right)} = \frac{\left(e^{pdg_{1464}} - e^{- pdg_{1464}}\right)^{2}}{\left(e^{pdg_{1464}} + e^{- pdg_{1464}}\right)^{2}}$$
valid 5349669879:
5349669879:
2121790783:
5349669879:
5349669879:
2121790783:
6 simplify
1. 8563535636; locally 4001109:
$$\cosh^2 x - \sinh^2 x = \left(\frac{\exp(x) + \exp(-x)}{2}\right)\left(\frac{\exp(x) + \exp(-x)}{2}\right) - \left(\frac{\exp(x) - \exp(-x)}{2}\right)\left(\frac{\exp(x) - \exp(-x)}{2}\right)$$
$$- \sinh^{2}{\left(pdg_{1464} \right)} + \cosh^{2}{\left(pdg_{1464} \right)} = \left(\frac{e^{pdg_{1464}}}{2} + \frac{e^{- pdg_{1464}}}{2}\right)^{2} - \frac{\left(e^{pdg_{1464}} - e^{- pdg_{1464}}\right)^{2}}{4}$$
1. 2762326680; locally 4009221:
$$\cosh^2 x - \sinh^2 x = \frac{1}{4}\left( \exp(2x)+1+1+\exp(-2x) - \left(\exp(2x)-1-1+\exp(-2x)\right) \right)$$
$$- \sinh^{2}{\left(pdg_{1464} \right)} + \cosh^{2}{\left(pdg_{1464} \right)} = 1$$
valid 8563535636:
2762326680:
8563535636:
2762326680:
9 declare initial expr
1. 8747785338; locally 7404421:
$$\cos(i x) = \cosh(x)$$
$$\cos{\left(pdg_{1464} pdg_{4621} \right)} = \cosh{\left(pdg_{1464} \right)}$$
no validation is available for declarations 8747785338:
8747785338:
5 subtract expr 1 from expr 2
1. 6031385191; locally 7844176:
$$\sinh^2 x = \left(\frac{\exp(x) - \exp(-x)}{2}\right)\left(\frac{\exp(x) - \exp(-x)}{2}\right)$$
$$\sinh^{2}{\left(pdg_{1464} \right)} = \left(\frac{e^{pdg_{1464}}}{2} - \frac{e^{- pdg_{1464}}}{2}\right)^{2}$$
2. 8532702080; locally 9245668:
$$\cosh^2 x = \left(\frac{\exp(x) + \exp(-x)}{2}\right)\left(\frac{\exp(x) + \exp(-x)}{2}\right)$$
$$\cosh^{2}{\left(pdg_{1464} \right)} = \left(\frac{e^{pdg_{1464}}}{2} + \frac{e^{- pdg_{1464}}}{2}\right)^{2}$$
1. 8563535636; locally 4001109:
$$\cosh^2 x - \sinh^2 x = \left(\frac{\exp(x) + \exp(-x)}{2}\right)\left(\frac{\exp(x) + \exp(-x)}{2}\right) - \left(\frac{\exp(x) - \exp(-x)}{2}\right)\left(\frac{\exp(x) - \exp(-x)}{2}\right)$$
$$- \sinh^{2}{\left(pdg_{1464} \right)} + \cosh^{2}{\left(pdg_{1464} \right)} = \left(\frac{e^{pdg_{1464}}}{2} + \frac{e^{- pdg_{1464}}}{2}\right)^{2} - \frac{\left(e^{pdg_{1464}} - e^{- pdg_{1464}}\right)^{2}}{4}$$
valid 6031385191:
8532702080:
8563535636:
6031385191:
8532702080:
8563535636:
4 multiply expr 1 by expr 2
1. 6404535647; locally 4319733:
$$\cosh x = \frac{\exp(x) + \exp(-x)}{2}$$
$$\cosh{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464}}}{2} + \frac{e^{- pdg_{1464}}}{2}$$
2. 6404535647; locally 4319733:
$$\cosh x = \frac{\exp(x) + \exp(-x)}{2}$$
$$\cosh{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464}}}{2} + \frac{e^{- pdg_{1464}}}{2}$$
1. 8532702080; locally 9245668:
$$\cosh^2 x = \left(\frac{\exp(x) + \exp(-x)}{2}\right)\left(\frac{\exp(x) + \exp(-x)}{2}\right)$$
$$\cosh^{2}{\left(pdg_{1464} \right)} = \left(\frac{e^{pdg_{1464}}}{2} + \frac{e^{- pdg_{1464}}}{2}\right)^{2}$$
valid 6404535647: error for dim with 6404535647
6404535647: error for dim with 6404535647
8532702080:
6404535647: N/A
6404535647: N/A
8532702080:
1 declare initial expr
1. 6404535647; locally 4319733:
$$\cosh x = \frac{\exp(x) + \exp(-x)}{2}$$
$$\cosh{\left(pdg_{1464} \right)} = \frac{e^{pdg_{1464}}}{2} + \frac{e^{- pdg_{1464}}}{2}$$
no validation is available for declarations 6404535647: error for dim with 6404535647
6404535647: N/A
Physics Derivation Graph: Steps for hyperbolic trigonometric identities

## Symbols for this derivation

$$i$$
$$x$$