# How can I obtain Wolfram Alpha's “alternative forms” of an expression inside Mathematica?

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.

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

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]}