Let $M$ be a smooth manifold, and let $K(M)$ denote its real $K$-theory ring, following Remark 2.7 at the nLab page.
Given a vector bundle $V$ we define the quantities $$ S_t(V) = \sum_{j=0}^{\infty} (S^jV)t^j \;\;,\;\; \Lambda_t(V) = \sum_{j=0}^{\infty} (\Lambda^jV)t^j $$ valued in $K(M)[[t]]$. (I assume $S_t(-V)$ and $\Lambda_t(-V)$ would be defined in the expected ways, by e.g. "absorbing the minus signs into $t$".)
I am trying to show that $S_t(V) \cdot \Lambda_{-t}(V) = 1 \in K(M)[[t]]$, for any $V$. (This is stated in e.g. the 3rd page, written as p. 31, in this paper.)
Following the suggestions in Prop 2.10 of the topological K-theory nLab page, I was able to reduce the above to showing that for any integer $\ell \geq 1$ and any (non-virtual) $V$ there is some other (non-virtual) vector bundle $F$ such that $$ F \oplus \left( \bigoplus_{\substack{0\leq j \leq \ell , \\ j\text{ even}}} S^{\ell-j}V \otimes \Lambda^{j}V \right) \simeq F \oplus \left( \bigoplus_{\substack{0\leq j \leq \ell , \\ j\text{ odd}}} S^{\ell-j}V \otimes \Lambda^{j}V \right) . $$
However here I am stuck. I was wondering if anyone could help explain why the above is always true, for some $F$? (Following the suggestions at the nLab page, if $M$ is compact we should be able to find such an $F$ which is also trivial.)