It is not true: the function is 2Pi-periodic, and bounded from above and below (and non-constant) so f'(x) must be negative for at least one positive value of x, which cannot be the case with only nonnegative coefficients of its series. (actually not being allowed to use a computer lead me to conjecture that it is not true ;) )
We have In[735]:= # == Normal[Series[#, {x, 0, 23}]] &[Tan[Sin[x]] - Sin[Tan[x]]]
Out[735]= -Sin[Tan[x]] + Tan[Sin[x]] == x^7/30 + 29 x^9/756 + 1913 x^11/75600 + 95 x^13/7392 + 311148869 x^15/54486432000 + 10193207 x^17/4358914560 + 1664108363 x^19/1905468364800 + 2097555460001 x^21/7602818775552000 + 374694625074883 x^23/6690480522485760000 Without using a computer, prove or disprove: The terms of the infinite series are nonnegative.
Christoph