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7

One way will be to add it to the definition of Cos by Unprotecting it. Unprotect[Cos] Cos[2 x] := 1 - 2 Sin[x]^2 Protect[Cos] Then evaluating the following: 2 Cos[2 x] + 3 x Cos[2 x] + Tan[x] Sin[x] Cos[2 x] + Exp[Tan[Cos[2 x]]] gives: Which you can further Simplify if you so please. Notice that the desired replacement has occurred everywhere ...

5

Extended comment: $Version "10.1.0 for Mac OS X x86 (64-bit) (March 24, 2015)" eq = 0.0055356626 + 0.000029 x + 0.000034 y <= 1000.; eq // Simplify x + 1.17241 y <= 3.44826*10^7 Use of assumptions reproduces the problem: Simplify[eq, {x > 0, y > 0}] False However, use of Rationalize avoids the problem with using ... 5 This bug has been fixed as of version 10.2.0. g[k2_] := (-510 k2 + (25761/4 - 6619 E^-k2) Log[26476/23721])/(-k2 + Log[26476/23721]) ans = Solve[g[k2] == 0, k2, Reals][[1, 1, 2]]; FullSimplify[ans] (* 8587/680 Log[26476/23721] + ProductLog[-(( 10579732023355105126853211737189792087286301059526347 (23721/ 6619)^(427/680) Log[26476/23721])/( ... 4 The OP correctly notes that Collect with Simplify is essential for computing this double sum in a reasonable amount of time, if at all. It is straightforward to find that the LeafCount of the inner sum only without using Collect is 721809. With it, the size of the inner sum drops to 21158. However, we can do much better. Instead, define the inner sum as ... 4 Often, it is desirable to apply Simplify or FullSimplify to parts of an expression and then recombine the parts. A particularly effective function for this purpose is Collect. For instance, Collect[expr, a^2 - b^2, Simplify] (* (4 a^2 - 3 b^2)/(a^2 - b^2)^2 - 8/(b^2 - c^2)^2 *) Although there is no need to use FullSimplifyin this case, it runs faster ... 4 It can be done with w.a.b.r + w.c.d.r /. Dot[FRONT_, AA__, BACK_] + Dot[FRONT_, BB__, BACK_] :> Dot[FRONT, Dot[AA] + Dot[BB], BACK] w.(a.b + c.d).r However, I like function argument destructuring, so I would probably write f[Dot[w_, a__, r_] + Dot[w_, b__, r_]] := w.(Dot[a] + Dot[b]).r f[w.a.b.r + w.c.d.r] w.(a.b + c.d).r 3 To answer the question as asked, we modify the code as follows: replacementRule = Plus[ Dot[FRONT__, AA__, BACK__] , Dot[FRONT__, BB__, BACK__] ] :> Dot[FRONT, Plus[Dot[AA], Dot[BB]], BACK]}] w.a.b.r + w.c.d.r /. replacementRule First, we have changed -> (Rule) to :> (RuleDelayed) so that when the expression is re-written, it will write it ... 3 Expanding Cos[2 x] to 1 - Sin[x]^2 and simplifying are somewhat opposite procedures. Simplify basically reduces the leaf count (see How do I invoke the default complexity function? for more details). As we can see the number of leaves in the goal is twice the numbers of leaves in the starting expression: LeafCount /@ {Cos[2 x], 1 - Sin[x]^2} (* {4, 8} ... 3 This appears to give you one approximate solution in a few seconds sol=NMinimize[ Norm[Abs[eta1]^2 + Abs[gamma1]^2 - 1] + Norm[Abs[eta2]^2 + Abs[gamma2]^2 - 1] + Norm[Abs[eta3]^2 + Abs[gamma3]^2 - 1] + Norm[Abs[tau1]^2 + Abs[kappa1]^2 - 1] + Norm[Abs[tau2]^2 + Abs[kappa2]^2 - 1] + Norm[Abs[tau3]^2 + Abs[kappa3]^2 -1 ] + Norm[Dot[q1, Conjugate[q1]] - ... 2 This is not a robust solution but since no one else was motivated to try to match your result here's my rough draft. expr1 = 225 a^4 + 600 a^3 b^2 + 520 a^2 b^4 + 160 a b^6 + 16 b^8 + 1200 a^4 b x + 2080 a^3 b^3 x + 960 a^2 b^5 x + 128 a b^7 x + 600 a^5 x^2 + 3120 a^4 b^2 x^2 + 2400 a^3 b^4 x^2 + 448 a^2 b^6 x^2 + 2080 a^5 b x^3 + 3200 a^4 b^3 x^3 + ... 2 e1 = 225 a^4 + 600 a^3 b^2 + 520 a^2 b^4 + 160 a b^6 + 16 b^8 + 1200 a^4 b x + 2080 a^3 b^3 x + 960 a^2 b^5 x + 128 a b^7 x + 600 a^5 x^2 + 3120 a^4 b^2 x^2 + 2400 a^3 b^4 x^2 + 448 a^2 b^6 x^2 + 2080 a^5 b x^3 + 3200 a^4 b^3 x^3 + 896 a^3 b^5 x^3 + 520 a^6 x^4 + 2400 a^5 b^2 x^4 + 1120 a^4 b^4 x^4 + 960 a^6 b x^5 + 896 a^5 b^3 x^5 + 160 ... 2 Here's some work-in-progress code I have been playing with. First we define a helper function which attempts to combine Anded comparisons (which result when using functions such as LogicalExpand) into Inequality, Equal and Unequal chains. It does this by making graphs of expressions, and finding paths connected components and cliques on them. It doesn't try ... 2 FullSimplify[] on the enormous expression gives this answer in less than a second:$(4 b^4 + 4 a^4 x^4 + 4 a b^2 (5 + 4 b x) + 4 a^3 x^2 (5 + 4 b x) + a^2 (15 + 8 b x (5 + 3 b x)))^2\$ It would take me several days to calculate such a result "by hand," and no doubt I would make errors.

2

You could go ahead and modify the mathematica output manually by defining exactly what you want to have transformed. In your case this would look like this: MakeBoxes[Cos[2*x_], StandardForm] := RowBox[{MakeBoxes[1 - 2 Sin[x]^2]}]; This effectively turns every occurrence of Cos[2x] in StandardForm into 1 - 2 Sin[x]^2. I use this to get a certain standard ...

2

A variant of the method from the other question: z = Sin[φ]*v; Dot[z, Conjugate[z]] (v Sin[φ]).Conjugate[v Sin[φ]] Simplify[%, φ ∈ Reals] //. Dot[a___, d_ b_ /; FreeQ[d, v], c___] :> d Dot[a, b, c] v.Conjugate[v] Sin[φ]^2 Integrate[%, {φ, 0, 2 π}] π v.Conjugate[v] Some explanation of the rule: //. -- "repeatedly apply the ...

1

It is not exactly what you asked about, but did you try this: list = {Cos[x], Sin[x], -(Cos[t3 (d1 - d2)]/(1728 tg^3 (d1 - d2)^2)) + Cos[t3 (d1 - d2)]/(864 tg^3 (d1 - d2) (-d1 + d2)) + Cos[t3 (d1 - d2)]/(864 tg^3 (d1 - d2) (2 (d1 - d2) - 2 (-d1 + d2))) - Cos[t3 (d1 - d2)]/(864 tg^3 (d1 - d2) (2 (d1 - d2) + 2 (-d1 + ...

1

(Caveat: I did the following experiments with paper, a pen, and Alpha, but I believe it should also work in Mathematica.) Through the use of a few well-known identities relating the incomplete gamma function and the partial sums of the exponential function, we have the following formula: f[n_Integer] := (n - 1) Sum[(n - k - 1)! Binomial[n - 1, n - k - 1], ...

1

expr = (b (1 - x)^n (-1 + x)^-n (-1 + x - w x)^n)/(n (-1 + w) (1 - x + w x)); Your stated assumptions including your comment assume = {Thread[Variables@Level[expr, {-1}] > 0], Element[n, Integers]} // Flatten {b > 0, n > 0, w > 0, x > 0, n \[Element] Integers} The simplified expression is expr // Simplify[#, assume] & (b (1 + (-1 + ...

1

Try using Simplify. x+(x*l) // Simplify

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