def.: $\displaystyle SO(3)$ is the collection of lineair orthogonal transformations of $\displaystyle \mathbb{R}^3$, the elements are $\displaystyle 3\times 3$ matrices.

I want to show that two element of $\displaystyle SO(3)$ are conjugated if and only if they have the same trace.

(Hint: Maby you want to use: if $\displaystyle A \in SO(3)\ \mbox{then}\ \exists p \in \mathbb{R}^3,\ p\neq 0\ \mbox{and}\ Ap=p $)