I'd like to show that for a topological space $X$, $$H_\ast(X\times S^n)\cong H_\ast(X)\oplus H_{\ast}(X\times S^n,X\times\{x_0\}).$$
It's obvious that the LES obtained from the SES of the pair has to be used, i.e. $$\rightarrow H_\ast(X)\overset{\iota_\ast}{\rightarrow} H_\ast(X\times S^n)\overset{\pi_\ast}{\rightarrow} H_\ast(X\times S^n,X\times\{x_0\})\rightarrow$$ with $\iota,\pi$ inclusion and projection, respectively. If this would split into SESs then the result would be obvious from the Splitting Lemma, however, I don't see why this should be the case.
I'd be grateful for any hints or explanations.