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Mathematica has tons of functions to perform algebraic manipulations. But is there a way to use it to check if I didn't make mistakes in my own algebraic manipulations.

As a real world example:

  • [step1] I have $\sin(\omega t + \pi /2)$
  • [step2] I rewrite that as $\sin(\omega t - \pi/4 +3\pi /4)$,
  • [step3] I apply a well known identity so I end up with $\sin(\omega t - \pi/4)\cos(3\pi /4) + \cos(\omega t - \pi/4)\sin(3\pi /4)$.

Will Mathematica be able to check: $$\text{step1}\Leftrightarrow\text{step2}\Leftrightarrow\text{step3}$$ I don't look for something smart, just verifying I didn't make a sign error or something like that.

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  • $\begingroup$ Instead of attempting it by hand, leave the Cos[a +b] in your expression, then use TrigExpand. E.g. TrigExpand@Cos[a + b] returns the identity you sought. See also TrigReduce etc. $\endgroup$
    – MarcoB
    Commented Dec 13, 2019 at 19:56
  • $\begingroup$ @Marco, I rewrote the question so it may eventually be more clear. For specific answer to you comment, in [step2], TrigExpand[Sin[\[Omega] t -a + b]] will perform a full expansion of the expression, which was not something I wanted. $\endgroup$
    – Sylvain Leroux
    Commented Dec 13, 2019 at 20:20
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    $\begingroup$ Simplify[step1-step3]? Or FullSimplify. When that doesn't work, I plug in random numbers with something like expr1 - expr2 /. x -> RandomReal[{-10, 10}, 1, WorkingPrecision -> 32], depending on what domain I want instead of {-10, 10}, how many variables there are, whether the expressions are listable. If listable, it's easy to check 100 random values with ... /. Thread[{x, y, z} -> RandomReal[{-2,2}, {3, 100}, WorkingPrecision -> 32]. $\endgroup$
    – Michael E2
    Commented Dec 13, 2019 at 20:34
  • $\begingroup$ Good idea @Michael. I was considering something like that but using == instead of a difference. I assume your solution is better suited to deal with rounding errors. $\endgroup$
    – Sylvain Leroux
    Commented Dec 13, 2019 at 22:27

1 Answer 1

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Quickly wrapping up Michael E2's idea:

s1 = Sin[w*t + \[Pi]/2];
s2 = Sin[w*t \[Minus] \[Pi]/4 + 3 \[Pi]/4];
s3 = Sin[w*t \[Minus] \[Pi]/4]*Cos[3 \[Pi]/4] + 
   Sin[3 \[Pi]/4]*Cos[w t - \[Pi]/4];
s4 = Sin[w*t + \[Pi]/2.1];

Clear[equal]; 
equal[s1_, s2_, n_ : 100, range_ : {-2, 2}] := Module[{
    vars = Quiet @ Union @ 
           Cases[s3, x_ /; ! ValueQ[x] && Context[x] == "Global`", {-1}, Heads -> True]},
  If[TrigExpand[s1] - TrigExpand[s2] == 0, True,
   tests = Thread[vars -> Table[Slot[i], {i, Length[vars]}]] & @@@ 
     RandomReal[range, {n, Length[vars]}, WorkingPrecision -> 32],
   Thread[s1 /. tests] == Thread[s2 /. tests]
   ]
  ]

equal @@@ {{s1, s2}, {s2, s3}, {s3, s4}}
(* {True, True, False} *)
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