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I was checking a trigonometric addition formula

cos(x)*cos(y) - sin(x)*sin(y) = cos(x+y)

using trigsimp() in SymPy.

from sympy import cos, sin, symbols, trigsimp

q = symbols('q')
print('trigsimp')
print(trigsimp(-sin(q)*sin(1*q) + cos(q)*cos(1*q)))
print(trigsimp(-sin(q)*sin(2*q) + cos(q)*cos(2*q)))
print(trigsimp(-sin(q)*sin(3*q) + cos(q)*cos(3*q)))
print(trigsimp(-sin(q)*sin(4*q) + cos(q)*cos(4*q)))
print(trigsimp(-sin(q)*sin(5*q) + cos(q)*cos(5*q)))
print(trigsimp(-sin(q)*sin(6*q) + cos(q)*cos(6*q)))

and the results are

trigsimp
cos(2*q)
cos(3*q)
cos(4*q)
-sin(q)*sin(4*q) + cos(q)*cos(4*q)
cos(6*q)
cos(7*q)

As shown above, the expression -sin(q)*sin(4*q) + cos(q)*cos(4*q)) was not simplified to cos(5*q).

I tried simplify() but the results were same.

Is this a type of pitfalls of the simplification in SymPy, or is there a way to simplify the expression above? I tried this condition with Python 3.8.5 and SymPy 1.11.1.

Thank you in advance for your help,

toyota Supra
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Kosuke Ohgo
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1 Answers1

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Fu, et al., developed an algorithm which applies transformations in a smart fashion to try to automatically come to a state a human work towards when simplifying a trigonometric expression. simplify and trigsimp use their own order of steps. In this case, fu is better:

from sympy import fu
>>> fu(-sin(q)*sin(4*q) + cos(q)*cos(4*q))
cos(5*q)

Beside the fu algorithm there is a dictionary with the individual "steps" in it that can be applied (or read about). In this case, TR8 and TR10i make the desired change:

>>> from sympy.simplify.fu import FU
>>> [i for i in FU if FU[i](-sin(q)*sin(4*q) + cos(q)*cos(4*q)) == cos(5*q)]
['TR8', 'TR10i']

>>> from sympy.simplify.fu import TR8
>>> [TR8(-sin(q)*sin(i*q) + cos(q)*cos(i*q)) for i in range(1,10)]
[cos(2*q), cos(3*q), cos(4*q), cos(5*q), cos(6*q), cos(7*q), cos(8*q), cos(9*q), cos(10*q)]
smichr
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  • Dear smichr, Thank you for your quick reply and providing detailed information. I will implement fu() to my actual code. Thank you again, Kosuke – Kosuke Ohgo Aug 16 '23 at 21:35
  • We have the reference for the fu algorithm in fu.py -- it was really an interesting read! – smichr Aug 17 '23 at 00:59
  • Thank you again, smichr ! It will be interesting to learn the deeper part of the algorithm. I am still a beginner of Python/SymPy. Also, I got used to use Symbolic Math tool box in MATLAB. I didn't have to think a lot to use that tool box. So I am realizing the difference of capabilities between SymPy and the Symbolic tool box. – Kosuke Ohgo Aug 17 '23 at 05:55
  • Any feedback you have about your experiences would be appreciated: https://groups.google.com/g/sympy – smichr Aug 17 '23 at 08:27