Prove that $\displaystyle \bold{u}_{r} \times (\bold{u}_{r} \times \bold{u}_{\theta}) = - \bold{u}_{\theta} $.
First read the start of this.
Then substitute $\displaystyle \vec{A} = \vec{B} = \bold{u}_{r}$ and $\displaystyle \vec{C} = \bold{u}_{\theta}$ into the formula.
Then note that $\displaystyle \bold{u}_{r}$ and $\displaystyle \bold{u}_{\theta}$ are perpendicular and simplify accordingly.