Answer
$$ \mathbf{T} =\cos t \mathbf{i}+\sin t \mathbf{j}$$
Work Step by Step
Since
$$\mathbf{r}=(\cos t+t \sin t) \mathbf{i}+(\sin t+t \cos t) \mathbf{j} $$
Then
\begin{align*}
\mathbf{v}&=(-\sin t+t \cos t+\sin t) \mathbf{i}+(\cos t-(t(-\sin t)+\cos t)) \mathbf{j}\\
&=(t \cos t) \mathbf{i}+(t \sin t) \mathbf{j} \\
|\mathbf{v}|&=\sqrt{(t \cos t)^{2}+(t \sin t)^{2}}&\\
&=\sqrt{t^{2}}=|t|=t,
\end{align*}
Hence
\begin{align*}
\mathbf{T}&=\frac{\mathbf{v}}{|\mathbf{v}|}\\
&=\frac{t \cos t}{t} \mathbf{i}+\frac{t \sin t}{t} \mathbf{j}\\
&=\cos t \mathbf{i}+\sin t \mathbf{j}\end{align*}
