I understand the concept that "radians" are a "dimensionless" angle measure. But, "radians" are an SI derived unit. There is a conversion to/from angular degrees. In practice, it is treated like a quantity. I think Mathematica is trying to accommodate unit based calculations by treating them as a Quantity. With that said...
One way to convert angular quantities to dimensionless "radians" expressions is to use a replacement rule like:
radrule = q_Quantity :> QuantityMagnitude[q, "Radians"]
Quantity[1, "Radians"] /. radrule
(* 1 *)
Quantity[1, "AngularDegrees"] /. radrule
(* \[Pi]/180 *)
1 /. radrule
(* 1 *)
Note: I've edited the code example above to show that the replacement rule, radrule
, leaves purely numeric expressions unchanged. I think this would be helpful for the 3 cases mentioned ("Radians", "AngularDegrees", or with no unit specified (dimensionless). In the dimensionless case, the assumption would be that the implied units are radians.
Regarding
QuantityMagnitude[Quantity[1, 1/"Seconds"], "Radians"/"Seconds"],
you can use the following to see the units are not compatible:
CompatibleUnitQ[1/"Seconds", "Radians"/"Seconds"]
(* False *)
I think this make sense because it would seem incorrect to assume that 1/"Seconds" can be unambiguously converted to "Radians"/"Seconds". However, the following is true:
CompatibleUnitQ[Quantity[1, "Hertz"], 1/"Seconds"]
(* True *)
The following links are a good starting point for working with units:
Symbolic Calculations with Units
Numerical Calculations with Units