Answer
$\textbf{T}(t) = \langle \frac{cos(t)}{\sqrt {1 + sin^2(t)}}, \frac{-sin(t)}{\sqrt {1 + sin^2(t)}}, \frac{-sin(t)}{\sqrt {1 + sin^2(t)}}\rangle$
Work Step by Step
$\textbf{r}(t) = \langle f(t), g(t), h(t)\rangle$
$\textbf{r}'(t) = \langle f'(t), g'(t), h'(t)\rangle$
$\textbf{r}(t) = \langle sin(t), cos(t), cos(t)\rangle$
$\textbf{r}'(t) = \langle cos(t), -sin(t), -sin(t)\rangle$
$|\textbf{r}'(t)| = \sqrt {(cos(t))^2 + (-sin(t))^2 + (-sin(t))^2} = \sqrt {1 + sin^2(t)}$
$\textbf{T}(t) = \frac{\textbf{r}'(t)}{|\textbf{r}'(t)|} = \frac{\langle cos(t), -sin(t), -sin(t)\rangle}{\sqrt {1 + sin^2(t)}} = \langle \frac{cos(t)}{\sqrt {1 + sin^2(t)}}, \frac{-sin(t)}{\sqrt {1 + sin^2(t)}}, \frac{-sin(t)}{\sqrt {1 + sin^2(t)}}\rangle$
