http://www.mapleprimes.com/posts/40158-Verifying-Identities en-us 2026 Maplesoft, A Division of Waterloo Maple Inc. Maplesoft Document System Fri, 12 Jun 2026 13:40:36 GMT Fri, 12 Jun 2026 13:40:36 GMT The latest comments added to the Post, Verifying identities http://www.mapleprimes.com/images/mapleprimeswhite.jpg http://www.mapleprimes.com/posts/40158-Verifying-Identities http://www.mapleprimes.com/posts/40158-Verifying-Identities?ref=Feed:MaplePrimes:Verifying identities:Comments#comment73859 This is not the type of problem Maple is designed to do, particularly the step-by-step illustration of the verification. Here is one way in which this identity can be "verified" with Maple. <pre> infolevel[all]:=10: sec(u + t) = (cos(u)*cos(t)+sin(u)*sin(t))/(cos(u)^2-sin(t)^2); cos(u) cos(t) + sin(u) sin(t) sec(u + t) = ----------------------------- 2 2 cos(u) - sin(t) simplify( lhs(%) ) = combine(rhs(%)); combine: combining with respect to trig combine: combining with respect to cmbtms combine: combining with respect to trig combine: combining with respect to trig combine: combining with respect to cmbpwr combine: combining with respect to power combine: combining with respect to power combine: combining with respect to power combine: combining with respect to power combine: combining with respect to radical combine: combining with respect to cmbpwr combine: combining with respect to power 1 2 cos(-u + t) ---------- = ------------------- cos(u + t) cos(2 u) + cos(2 t) %*denom(lhs(%)); 2 cos(u + t) cos(-u + t) 1 = ------------------------ cos(2 u) + cos(2 t) lhs(%) = combine( rhs(%) ); combine: combining with respect to trig combine: combining with respect to cmbtms combine: combining with respect to cmbpwr combine: combining with respect to power 1 = 1 </pre> The infolevel command tells Maple to give us a little more information about what it is doing (or trying). In this case, we only get minimal information from combine. It would be nice to get a little more information. I'm sure others will have more suggestions. Hopefully, at least one of our suggestions will be useful - even if it motivates you to write the procedure with the specific functionality that you seek. Doug <pre> --------------------------------------------------------------------- Douglas B. Meade Math, USC, Columbia, SC 29208 E-mail: mailto:meade@math.sc.edu Phone: (803) 777-6183 URL: http://www.math.sc.edu/~meade/ </pre> The latest comments added to the Post, Verifying identities 73859 Tue, 29 Jan 2008 08:35:50 Z Doug Meade Doug Meade http://www.mapleprimes.com/posts/40158-Verifying-Identities?ref=Feed:MaplePrimes:Verifying identities:Comments#comment73858 To what extent is the verification of trigonometrical identities limited by the undecidability of the zero equivalence? The latest comments added to the Post, Verifying identities 73858 Tue, 29 Jan 2008 10:19:18 Z jakubi jakubi http://www.mapleprimes.com/posts/40158-Verifying-Identities?ref=Feed:MaplePrimes:Verifying identities:Comments#comment84333 I hope I have this right. Richardson's undecidability result, as sharpened by Caviness, shows there is no algorithm that, given a function G(x) of one variable constructed using rational numbers, pi, addition, multiplication, composition and the sin function, decides whether there exists real x such that G(x) < 0. Thus there is no way to verify whether sqrt(G(x)^2) = G(x) is a correct identity [using the positive square root]. The latest comments added to the Post, Verifying identities 84333 Tue, 29 Jan 2008 10:56:45 Z Robert Israel Robert Israel http://www.mapleprimes.com/posts/40158-Verifying-Identities?ref=Feed:MaplePrimes:Verifying identities:Comments#comment92620 that at least Richardson's results also required the abs function to be in that set (at least <a href="http://mathworld.wolfram.com/RichardsonsTheorem.html">according to MathWorld</a>). Plus, it doesn't say that it's undecidable, but that it is recursively undecidable. Frequently a semi-decision procedure is much more useful that a total predicate [and hence true/false/FAIL in Maple]. Note that this undecidability result only says that <em>arbitrary</em> identities cannot be checked, it does not say that there are not large sub-classes of expressions for which zero-equivalence is decidable. So undecidability results always have to be taken with a large grain of salt, since they rarely stated in 'effective' terms (ie all expressions that contain all these symbols and larger than a specific side are all undecidable, for example; the results are all existential). I believe that buried in one of <a href="http://www.math.u-psud.fr/~vdhoeven/">Joris van der Hoeven's</a> papers is an algorithm which would indeed decide trigonometric identities such as the ones the original poster asked about. Richardson's trick to get undecidability was to show how to 'encode' tough diophantine problems (already shown undecidable) into a zero-decision problem for the class of functions in the MathWorld article. The point is that it is not a 'new' undecidability result, but really an encoding of an old one in a new guise. If you stick to trig functions and rational coefficients, I do believe the class is quite well-behaved. The latest comments added to the Post, Verifying identities 92620 Wed, 30 Jan 2008 04:42:34 Z JacquesC JacquesC http://www.mapleprimes.com/posts/40158-Verifying-Identities?ref=Feed:MaplePrimes:Verifying identities:Comments#comment73843 In a more helpful vein, if you have a particular set of rewrite rules for trigonometrics, then it is quite easy to implement these in Maple. Then you can implement a proof search routine on top of that, following whatever heuristics you like. If your search finds an identity, then you can simply look at the path taken by the search to find your proof. This is more-or-less what some simplistic automated theorem proving systems do [and not at all what the advanced ones do]. But that should be sufficient for trig identities. The latest comments added to the Post, Verifying identities 73843 Wed, 30 Jan 2008 04:47:11 Z JacquesC JacquesC http://www.mapleprimes.com/posts/40158-Verifying-Identities?ref=Feed:MaplePrimes:Verifying identities:Comments#comment92621 Now I'm confused (this does not imply that I wasn't confused before, just that I now know that I'm confused). I based my statement on <a href="http://portal.acm.org/citation.cfm?doid=321850.321856"> Paul Wang's article</a> which doesn't mention the absolute value function when it refers to "Caviness' improved version of Richardson's Theorem", but skimming through <a href="http://portal.acm.org/citation.cfm?id=321591">Caviness's article</a> it looks like the absolute value is required after all for that result. The latest comments added to the Post, Verifying identities 92621 Wed, 30 Jan 2008 05:43:49 Z Robert Israel Robert Israel