Subspace codes, especially cyclic constant subspace codes, are of great use in random network coding. Subspace codes can be constructed by subspaces and subspace polynomials. In particular, many researchers are keen to find special subspaces and subspace polynomials to construct subspace codes with the size and the minimum distance as large as possible. In [14], Roth, Raviv and Tamo constructed several subspace codes using Sidon spaces, and it is proved that subspace codes constructed by Sidon spaces has the largest size and minimum distance. In [12], Niu, Yue and Wu extended some results of [14] and obtained several new subspace codes. In this paper, we first provide a sufficient condition for the sum of Sidon spaces is again a Sidon space. Based on this result, we obtain new cyclic constant subspace codes through the sum of two and three Sidon spaces. Our results generalize the results in [14] and [12].