Stability of vector bundles (Ex)

Let $E \to X$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}E \to X$ and $F \to X$$F \to X$ be rank $k$$k$ vector bundles over an $n$$n$-dimensional CW-complex $X$$X$ with $k > n$$k > n$.

1) Prove that $E$$E$ is isomorphic to $F$$F$ if and only if $E$$E$ is stably isomorphic to $F$$F$; i.e. there is an isomorphism $E \oplus \underline{\R^i} \cong F \oplus \underline{\R^i}$$E \oplus \underline{\R^i} \cong F \oplus \underline{\R^i}$ for some $i \geq 0$$i \geq 0$, where $\underline{\R^i}$$\underline{\R^i}$ denotes the trivial rank $i$$i$ bundle over $X$$X$.

2) If $k > n+1$$k > n+1$, prove that bundle isomorphisms $\theta_1, \theta_2 \colon F \to E$$\theta_1, \theta_2 \colon F \to E$ are fibrewise homotopic if and only if $\theta_1$$\theta_1$ and $\theta_2$$\theta_2$ are stably homotopic; i.e. there is a fibrewise homotopy between $\theta_1 \oplus \mathrm{id}_{\underline{\R^i}}$$\theta_1 \oplus \mathrm{id}_{\underline{\R^i}}$ and $\theta_1 \oplus \mathrm{id}_{\underline{\R^i}}$$\theta_1 \oplus \mathrm{id}_{\underline{\R^i}}$ for some $i \geq 0$$i \geq 0$.