Question: Show that two vector bundles $HTM$ and $\pi^*T^*M$ are isomorphic as vector bundles over $TM$.
Consider $$\pi:TM\to M \quad; \quad (x,y)\to x,$$ $$\pi_{TM}:TTM= HTM \oplus VTM\to TM\quad ; \quad (x,y, X, Y)\to (x,y),$$ $$\pi_*: TTM \to TM\quad ; \quad (x,y, X, Y)\to (x,X),$$ $$\tau:T^*M \to M \quad , \quad (x, \alpha)\to x,$$ and pull_ back bundle $$\pi^*T^*M\subset TM\times T^*M\to TM\quad, \quad \pi^*T^*M := \{(v, \alpha)| \pi(u)=\tau(\alpha)\}\to v.$$ I know $\pi^{*}TM\simeq VTM$ and $\pi ^{*}TM \simeq \pi^*T^*M$, then $VTM \simeq \pi^*T^*M$. But I do not know how to prove $HTM \simeq \pi^*T^*M$.