K theory of suspension of the disjoint union

Question

I wonder why it is true that $\widetilde{K^0}(S(X \sqcup \{*\}))=\widetilde{K^0}(SX) \oplus \widetilde{K^0}(S^1)$ where $SX$ is the suspension of $X$ and $\widetilde{K^0}$ is reduced $K^0$ group.