The set of vectors $\{\sinh(x),\cosh(x),e^{x}\}$ is a linearly dependent set because the last vector (function) can be written as a linear combination of the two previous ones, more specifically as the sum of the previous two.
Hence if we remove it and consider only the functions spanned by $\{\sinh(x),\cosh(x)\}$ we get the same subspace!
This is true in general. If you have a linear dependent set of vectors, you can always remove a vector that is in the span of the others, and the new (smaller) set of vectors will generate the same subspace.