In these lecture notes (https://arxiv.org/abs/1506.01961) on composite Higgs models, on page 22, the authors calculate the Goldstone Matrix $U[\Pi]$ for an abelian composite Higgs scenario i.e. $SO(3)$ group breaks to $SO(2)$. The matrix $U[\Pi]$ is defined as \begin{equation} U[\Pi] = \text{exp}\left(i \frac{\sqrt{2}}{f}\Pi^{i}(x)\hat{T}_i\right), \end{equation} where the $\hat{T}_i$ are the broken generators of the $SO(3)$ group, given by $$T_1=\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & 0 & -i \\ 0 & 0 & 0 \\ i & 0 & 0 \\ \end{pmatrix}$$ $$T_2=\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \\ \end{pmatrix}$$ and $\Pi_1$, $\Pi_2$ are the goldstone fields associated with the breaking.
The authors give the form of $U[\Pi]$ matrix as, $$U[\Pi] = \text{exp}\left(i \frac{\sqrt{2}}{f}\Pi^{i}(x)\hat{T}_i\right) = \begin{pmatrix} \mathbb{I} - \left( 1- \cos \frac{\Pi}{f}\right) \frac{\vec{\Pi}\:\vec{\Pi}^\intercal}{\Pi^2} & \sin \frac{\Pi}{f} \frac{\vec{\Pi}}{\Pi} \\ -\sin \frac{\Pi}{f} \frac{\vec{\Pi}^{\intercal}}{\Pi} & \cos \frac{\Pi}{f}\end{pmatrix}$$ where $\Pi = \sqrt{\vec{\Pi}^{\intercal}\vec{\Pi}}$.
From the definition of the matrix $U[\Pi]$ getting to this form is not clear to me, particularly the $\cos \frac{\Pi}{f}$ and $\sin \frac{\Pi}{f}$ terms. I tried expanding the exponential (up to order 3) and separated the sin (odd) and cosine (even) terms, but it does not look anything like the form mentioned in the final equation above. Please help.
