Actually, I don't know. It's like using a trig function just triggers a rewrite in terms of trig functions whereas requesting `tan` does not:

```
>>> (exp(x)+exp(I*x)).rewrite(cos)
I*sin(x) + cos(x) + sinh(x) + cosh(x)
>>> (exp(x)+exp(I*x)).rewrite(sin)
I*sin(x) + cos(x) + sinh(x) + cosh(x)
>>> (exp(x)+exp(I*x)).rewrite(tan)
exp(x) + exp(I*x)
```