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Question: How to transform this sine expression into a more simpler form for small 𝑚?

Question:How to transform this sine expression into a more simpler form for small 𝑚?

sursumCorda 1099 Maple 2024
simplify simplification trigonometric combine
March 18 2024
1

I am trying to get Maple to simplify the following trigonometric expressions (for "generic" parameters) as much as possible: 

sineExpr(3) := (
   sin(a[2] - b[1])*sin(a[3] - b[1]))/(
   sin(b[2] - b[1])*sin(b[3] - b[1]))*sin(a[1] - b[1]) + (
   sin(a[3] - b[2])*sin(a[1] - b[2]))/(
   sin(b[3] - b[2])*sin(b[1] - b[2]))*sin(a[2] - b[2]) + (
   sin(a[1] - b[3])*sin(a[2] - b[3]))/(
   sin(b[1] - b[3])*sin(b[2] - b[3]))*sin(a[3] - b[3]);
 = 
   (('RealDomain:-sin'(a[2]-b[1])*'RealDomain:-sin'(a[3]-b[1]))/('RealDomain:-sin'(b[2]-b[1])*'RealDomain:-sin'(b[3]-b[1])))&*'RealDomain:-sin'(a[1]-b[1])+(('RealDomain:-sin'(a[3]-b[2])*'RealDomain:-sin'(a[1]-b[2]))/('RealDomain:-sin'(b[3]-b[2])*'RealDomain:-sin'(b[1]-b[2])))&*'RealDomain:-sin'(a[2]-b[2])+(('RealDomain:-sin'(a[1]-b[3])*'RealDomain:-sin'(a[2]-b[3]))/('RealDomain:-sin'(b[1]-b[3])*'RealDomain:-sin'(b[2]-b[3])))&*'RealDomain:-sin'(a[3]-b[3])


sineExpr(4) := (
   sin(a[2] - b[1])*sin(a[3] - b[1])*sin(a[4] - b[1]))/(
   sin(b[2] - b[1])*sin(b[3] - b[1])*sin(b[4] - b[1]))*
   sin(a[1] - b[1]) + (
   sin(a[3] - b[2])*sin(a[4] - b[2])*sin(a[1] - b[2]))/(
   sin(b[3] - b[2])*sin(b[4] - b[2])*sin(b[1] - b[2]))*
   sin(a[2] - b[2]) + (
   sin(a[4] - b[3])*sin(a[1] - b[3])*sin(a[2] - b[3]))/(
   sin(b[4] - b[3])*sin(b[1] - b[3])*sin(b[2] - b[3]))*
   sin(a[3] - b[3]) + (
   sin(a[1] - b[4])*sin(a[2] - b[4])*sin(a[3] - b[4]))/(
   sin(b[1] - b[4])*sin(b[2] - b[4])*sin(b[3] - b[4]))*
   sin(a[4] - b[4]);
 = 
(('RealDomain:-sin'(a[2]-b[1])*'RealDomain:-sin'(a[3]-b[1])*'RealDomain:-sin'(a[4]-b[1]))/('RealDomain:-sin'(b[2]-b[1])*'RealDomain:-sin'(b[3]-b[1])*'RealDomain:-sin'(b[4]-b[1])))&*'RealDomain:-sin'(a[1]-b[1])+(('RealDomain:-sin'(a[3]-b[2])*'RealDomain:-sin'(a[4]-b[2])*'RealDomain:-sin'(a[1]-b[2]))/('RealDomain:-sin'(b[3]-b[2])*'RealDomain:-sin'(b[4]-b[2])*'RealDomain:-sin'(b[1]-b[2])))&*'RealDomain:-sin'(a[2]-b[2])+(('RealDomain:-sin'(a[4]-b[3])*'RealDomain:-sin'(a[1]-b[3])*'RealDomain:-sin'(a[2]-b[3]))/('RealDomain:-sin'(b[4]-b[3])*'RealDomain:-sin'(b[1]-b[3])*'RealDomain:-sin'(b[2]-b[3])))&*'RealDomain:-sin'(a[3]-b[3])+(('RealDomain:-sin'(a[1]-b[4])*'RealDomain:-sin'(a[2]-b[4])*'RealDomain:-sin'(a[3]-b[4]))/('RealDomain:-sin'(b[1]-b[4])*'RealDomain:-sin'(b[2]-b[4])*'RealDomain:-sin'(b[3]-b[4])))&*'RealDomain:-sin'(a[4]-b[4])

So far, all of my attempts have failed: 
 

> 

restart:
> 

kernelopts('version');
> 

Physics:-Version();

`Maple 2024.0, X86 64 WINDOWS, Mar 01 2024, Build ID 1794891`

  

`The "Physics Updates" version in the MapleCloud is 1701 and is the same as the version installed in this computer, created 2024, March 17, 17:24 hours Pacific Time.`

 (1)
> 

sineExpr := proc (m::posint) options operator, arrow; add(mul(ifelse(j <> t, (':-sin')(a[j]-b[t])/(':-sin')(b[j]-b[t]), (':-sin')(a[t]-b[t])), j = 1 .. m), t = 1 .. m) end proc

Warning, (in sineExpr) `t` is implicitly declared local

  

Warning, (in sineExpr) `j` is implicitly declared local

  

Warning, (in sineExpr) `t` is implicitly declared local

  

Warning, (in sineExpr) `j` is implicitly declared local

  
> 

combine(simplify(normal(sineExpr(1), expanded), trig), trig);

sin(a[1]-b[1])

 (2)
> 

combine(simplify(normal(sineExpr(2), expanded), trig), trig); # which can be transformed into sin((a[1]+a[2])-(b[1]+b[2])) only in certain legacy versions!

-(1/2)*(cos(-2*b[2]+a[1]+a[2])-cos(-2*b[1]+a[1]+a[2]))/sin(b[1]-b[2])

 (3)
> 

combine(simplify(normal(sineExpr(3), expanded), trig), trig);

(1/2)*(cos(-b[1]-3*b[2]+b[3]+a[2]+a[3]+a[1])-cos(b[1]-3*b[2]-b[3]+a[2]+a[3]+a[1])-cos(-b[1]-3*b[3]+a[2]+a[3]+a[1]+b[2])+cos(b[1]-3*b[3]-b[2]+a[2]+a[3]+a[1])+cos(-3*b[1]+a[2]+a[3]+a[1]+b[2]-b[3])-cos(-3*b[1]+a[2]+a[3]+a[1]-b[2]+b[3]))/(sin(-2*b[2]+2*b[1])-sin(-2*b[3]+2*b[1])+sin(2*b[2]-2*b[3]))

 (4)
> 

CodeTools:-Usage(combine(simplify(normal(sineExpr(4), expanded), trig), trig));

memory used=244.67MiB, alloc change=0 bytes, cpu time=6.17s, real time=5.49s, gc time=1000.00ms

  

(1/2)*(sin(-2*a[3]+4*a[4]+2*a[2]-b[3]-b[4]-b[2]-b[1])-sin(4*a[2]-2*a[4]-b[3]-b[4]-b[2]-b[1]+2*a[1])+sin(-2*a[2]-b[3]-b[4]-b[2]-b[1]+4*a[4]+2*a[1])+sin(-2*a[2]+b[3]+b[4]+b[2]+b[1]-4*a[4]+2*a[1])+sin(-b[3]-b[4]-b[2]-b[1]+4*a[1]-2*a[3]+2*a[4])-sin(4*a[3]-2*a[4]+2*a[2]-b[3]-b[4]-b[2]-b[1])-sin(-4*a[3]-2*a[4]+2*a[2]+b[3]+b[4]+b[2]+b[1])+sin(-2*a[3]-4*a[4]+2*a[2]+b[3]+b[4]+b[2]+b[1])-sin(-2*a[3]+2*a[4]+4*a[2]-b[3]-b[4]-b[2]-b[1])+sin(-4*a[3]+b[3]+b[4]+b[2]+b[1]-2*a[4]+2*a[1])-sin(4*a[4]-2*a[3]-b[3]-b[4]-b[2]-b[1]+2*a[1])-sin(-2*a[2]-b[3]-b[4]-b[2]-b[1]+4*a[3]+2*a[1])-sin(-2*a[2]+b[3]+b[4]+b[2]+b[1]-4*a[3]+2*a[1])+sin(2*a[3]-2*a[4]+4*a[2]-b[3]-b[4]-b[2]-b[1])-sin(-4*a[2]-2*a[4]+b[3]+b[4]+b[2]+b[1]+2*a[1])+sin(4*a[3]-b[3]-b[4]-b[2]-b[1]-2*a[4]+2*a[1])-sin(-b[3]-b[4]-b[2]-b[1]+4*a[1]+2*a[3]-2*a[4])-sin(-b[3]-b[4]-b[2]-b[1]+4*a[1]+2*a[4]-2*a[2])+sin(-b[3]-b[4]-b[2]-b[1]+4*a[1]-2*a[4]+2*a[2])+sin(-b[3]-b[4]-b[2]-b[1]+4*a[1]+2*a[3]-2*a[2])-sin(-b[3]-b[4]-b[2]-b[1]+4*a[1]-2*a[3]+2*a[2])-sin(-4*a[4]-2*a[3]+b[3]+b[4]+b[2]+b[1]+2*a[1])+sin(4*a[2]-2*a[3]-b[3]-b[4]-b[2]-b[1]+2*a[1])+sin(-4*a[2]-2*a[3]+b[3]+b[4]+b[2]+b[1]+2*a[1]))/(cos(a[1]-a[2]-3*a[3]+3*a[4])-cos(a[1]-a[2]+3*a[3]-3*a[4])+cos(-3*a[2]-a[3]+a[4]+3*a[1])-cos(-3*a[2]+a[3]-a[4]+3*a[1])-cos(a[1]+3*a[2]-3*a[3]-a[4])+cos(a[3]-3*a[4]-a[2]+3*a[1])-cos(-a[3]-3*a[4]+a[2]+3*a[1])+cos(-3*a[3]-a[4]+a[2]+3*a[1])-cos(-3*a[3]+a[4]-a[2]+3*a[1])+cos(a[1]-3*a[2]+3*a[3]-a[4])-cos(a[1]-3*a[2]-a[3]+3*a[4])+cos(a[1]+3*a[2]-a[3]-3*a[4]))

 (5)
> 

CodeTools:-Usage(combine(simplify(normal(sineExpr(5), expanded), trig), trig)): # rather lengthy

memory used=4.23GiB, alloc change=-32.00MiB, cpu time=2.66m, real time=2.29m, gc time=29.98s

  

Can sineExpr(3), sineExpr(4), and sineExpr(5) be reduced to sin(a[1]+a[2]+a[3]-b[1]-b[2]-b[3]), sin(a[1]+a[2]+a[3]+a[4]-b[1]-b[2]-b[3]-b[4]), and sin(a[1]+a[2]+a[3]+a[4]+a[5]-b[1]-b[2]-b[3]-b[4]-b[5]) respectively by Maple itself (that is, with as little user-intervention as possible) if one is not aware of such reductions in advance?


 

Download sinIdentity.mws

Note that because zero testing is frequently considerably easier, combine always succeeds in showing that the difference between the simplest possible and the original version is zero. 

combine(sin((a[1]+a[2]+a[3])-(b[1]+b[2]+b[3]))-sineExpr(3));
 = 
                               0

combine(sin((a[1]+a[2]+a[3]+a[4])-(b[1]+b[2]+b[3]+b[4]))-sineExpr(4));
 = 
                               0

However, I wonder if Maple can thoroughly simplify them without knowing those known “simplest possible” form beforehand. 
I also tried some other functions like rationalize, radnormal, and `convert/trig`, yet Maple appears to have not been capable of completely simplifying even the sub-simplest case 𝑚＝2. Is there any workaround? 

Of note, it can be demonstrated inductively that ∀m∈ℕ₊, 

 

where none of the denominators is 0. Nevertheless, as mentioned above, is it possible to transform sineExpr(3) and sineExpr(4) (as well as sineExpr(5), if possible) into potentially more elegant forms (Ideally, sineExpr(3) is rewritten into sin((a1＋a2＋a3)－(b1＋b2＋b3)), sineExpr(4) is rewritten into sin((a1＋a2＋a3＋a4)－(b1＋b2＋b3＋b4)), and sineExpr(5) is rewritten into sin((a1＋a2＋a3＋a4＋a5)－(b1＋b2＋b3＋b4＋b5)).) without any such a priori knowledge? 
In Mma, these may be done using TrigReduce directly (cf. ); unfortunately, I cannot found a Maple equivalent to such functionality. 
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