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When I type in Wolfram Alpha, say,

cos(a+b)

or

cos(a+b)^2

or (OK this next one can easily be obtained in Mathematica, still, I have run into Bessel function expressions that cannot be simplified by Mathematica, but get a decent "alternative expression" through W|A)

D[A^mu BesselK[mu, A r], A]

I get several alternative forms of these, but I cannot seem to obtain them simply from pure Mathematica (i.e. no W|A integration stuff). Am I missing something or is Wolfram Alpha just smarter in these things? The alternative expressions have helped me quite a lot, so it would be nice to have them in my toolbox without having to access Alpha.

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  • $\begingroup$ Where do you type those? $\endgroup$ – Yves Klett Jun 12 '14 at 13:10
  • $\begingroup$ @YvesKlett In W|A I think $\endgroup$ – Öskå Jun 12 '14 at 13:11
  • $\begingroup$ @Öskå yes, indeed, missed that when writing it down :-) $\endgroup$ – rubenvb Jun 12 '14 at 13:12
  • $\begingroup$ You can query W|A by entering = or ==? $\endgroup$ – Yves Klett Jun 12 '14 at 13:12
  • $\begingroup$ So @rubenvb Check #@Cos[a+b]&/@{TrigToExp,TrigExpand} $\endgroup$ – Öskå Jun 12 '14 at 13:19
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You can call Wolfram Alpha directly from the notebook,

 Part[#, 2] & /@ 
       WolframAlpha[
        "cos(a+b)^2",{{"AlternativeRepresentations:MathematicalFunctionIdentityData", All},
        "Content"},PodStates ->{"AlternativeRepresentations:MathematicalFunctionIdentityData__More"}]

it should give you all the alternate forms.

 {HoldForm[Cos[a + b]^2 == Cosh[(a + b)*I]^2], 
  HoldForm[Cos[a + b]^2 == (1/Sec[a + b])^2], 
  HoldForm[Cos[a + b]^2 == Cosh[(-I)*(a + b)]^2], 
  HoldForm[Cos[a + b]^2 == (1/Csc[a + b + Pi/2])^2], 
  HoldForm[Cos[a + b]^2 == ((1/2)*(E^((-I)*(a + b)) + E^((a + b)*I)))^2], 
  HoldForm[Cos[a + b]^2 == (1/Csc[-a - b + Pi/2])^2], 
  HoldForm[Cos[a + b]^2 == (-(I/Csch[(a + b)*I + (I*Pi)/2]))^2], 
  HoldForm[Cos[a + b]^2 == (-(I/Csch[-((a + b)*I) + (I*Pi)/2]))^2]}
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