Why does TrigExpand[Cos[a+b]] return the familiar and expected
Cos[a] Cos[b] - Sin[a] Sin[b]
but TrigExpand[Cos[2 Pi n x + b]] return
Cos[b] Cos[n \[Pi] x]^2 - 2 Cos[n \[Pi] x] Sin[b] Sin[n \[Pi] x] - Cos[b] Sin[n \[Pi] x]^2?
I would have thought that Mathematica would group the terms into a single constant.
I can of course perform TrigExpand[Cos[a+b]] /. a -> 2 Pi n x,
but that seems rather inelegant.
It is because of the 2. Compare
TrigExpand[Cos[a + b]] 
With
TrigExpand[Cos[2 a + b]] 
This is because
TrigExpand[Cos[2 a]] gives
Using your original expression, if you remove the 2 now you get what you expected
TrigExpand[Cos[Pi n x + b]] 
Having the 2 in there made it use that intermediate rule which caused the output to look completely different now. There is probably a way to turn off that rule for Cos[2 x] from expanding.
At the risk of breaking something internal, you can modify TrigReduce to change only the case when Cos[2 x] is in play:
Unprotect[TrigExpand]; TrigExpand[Cos[2*x_ + y_]] := Cos[2*x] Cos[y] - Sin[2*x] Sin[y] Protect[TrigExpand]; and now
TrigExpand[Cos[2 Pi n x + b]] 
TrigExpand[Cos[2. Pi n x + b]] (note the 2.). Thanks ($\checkmark$). $\endgroup$
Detailssection:TrigExpand splits up sums and integer multiples that appear in arguments of trigonometric functions, and then expands out products of trigonometric functions into sums of powers, using trigonometric identities when possible.$\endgroup$