Physics Derivation Graph navigation Sign in

review derivation: hyperbolic trigonometric identities

This page contains three views of the steps in the derivation: d3js, graphviz PNG, and a table.


Hold the mouse over a node to highlight that node and its neighbors. You can zoom in/out. You can pan the image. You can move nodes by clicking and dragging.

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

Options
Alternate views of this derivation:
Edit this content:    

To edit a step, click on the number in the "Index" column in the table below

Clicking on the step index will take you to the page where you can edit that step.

Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
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:
6404535647:
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:
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:
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:
6404535647:
8532702080:
6404535647:
6404535647:
8532702080:
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:
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:
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:
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:
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:
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:
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:
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:
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:
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:
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:
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:
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:
8747785338:
8651044341:
6404535647:
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:
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:
7731226616:
4166155526:
6404535647:
7731226616:
4166155526:
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:
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:
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:
2902772962:
5349669879:
6404535647:
2902772962:
5349669879:
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:
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:
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:
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:
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: error for dim with 5866629429
4830221561:
5866629429: N/A
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: error for dim with 5866629429
5866629429: N/A
Physics Derivation Graph: Steps for hyperbolic trigonometric identities

Symbols for this derivation

See also all 212 symbols
symbol ID category latex scope dimension name value Used in derivations references
1464 variable x
\(x\)
['real']
140
4621 variable i
\(i\)
['imaginary']
imaginary unit 74
MESSAGE: