tan(x) and sin(x) are equal numbers in a calculator for small enough x because the calculator does not represent enough digits. Hence the numerator turns to 0.
In the Windows calculator, your example works fine, but:
sin(0.0000000000000001) = tan(0.0000000000000001) = 1.7453292519943295769236907684886e-18
The difference is in the mantissa (the first part), which cannot properly represent such tiny sines and tangents. It needs more digits because that's where the difference between sine and tangent show up with such tiny x. This would be important in any formula that relied on those functions with such tiny values -- the output of the formula would be junk. After all, if sin(x) = tan(x) for every tiny number, well, now what?
Note that it will happily cube the sin, though. 1.7453292519943295769236907684886e-18 cubed is
And 0 divided by that is still 0 (the original problem result). Note sin cubed of a junk sin value (also equal to tan) is also meaningless. It's the mantissa (the first part) not the exponent (the second part) that has strayed into meaninglessness.