Answer
$cos\;\theta$
Work Step by Step
By using the cofactor expansion along the first row:
$det\begin{vmatrix}
cos\;\theta&sin\;\theta &tan\;\theta\\
0&cos\;\theta &-sin\;\theta\\
0&sin\;\theta&cos\;\theta
\end{vmatrix}$
$=+cos\;\theta\times det\begin{vmatrix}
cos\;\theta&-sin\;\theta\\
sin\;\theta&cos\;\theta
\end{vmatrix}-sin\;\theta\times det\begin{vmatrix}
0 &-sin\;\theta\\
0&cos\;\theta
\end{vmatrix}+tan\;\theta\times det\begin{vmatrix}
0&cos\;\theta\\
0&sin\;\theta
\end{vmatrix}$
$=cos\;\theta(cos^2\;\theta+sin^2\;\theta)-sin\;\theta(0-0)+tan\;\theta(0-0)$
Recall that:$\;\{ sin^2\;\theta+cos^2\;\theta=1\}$
$=cos\;\theta(1)-0+0$
$=cos\;\theta$
