Answer
See below
Work Step by Step
According to Cramer's Rule for a $2\times 2$ system $Ax=b$ where $A=\begin{bmatrix}
a_1 &b_1\\a_2&b_2
\end{bmatrix}$ and $b=\begin{bmatrix}
c_1\\c_2
\end{bmatrix}$
we have $x_1=\frac{\begin{vmatrix}
c_1 &b_1\\c_2&b_2
\end{vmatrix}}{\begin{vmatrix}
a_1 &b_1\\a_2&b_2
\end{vmatrix}}=\frac{\begin{vmatrix}
e^{-t} &\sin t\\3e^{-t}&-\cos t
\end{vmatrix}}{\begin{vmatrix}
\cos t &\sin t\\ \sin t& -\cos t
\end{vmatrix}}=\frac{\frac{-\cos t}{e^t}-\frac{3\sin t}{e^t}}{-\cos^2(t)-\sin^2 (t)}=\frac{\cos t+3\sin t}{e^t}$
and $x_2=\frac{\begin{vmatrix}
a_1 &c_1\\a_2&c_2
\end{vmatrix}}{\begin{vmatrix}
a_1 &b_1\\a_2&b_2
\end{vmatrix}}=\frac{\begin{vmatrix}
\cos t&e^{-t}\\\sin t& 3e^{-t}
\end{vmatrix}}{\begin{vmatrix}
\cos t& \sin t\\ \sin t& -\cos t
\end{vmatrix}}=\frac{\frac{3\cos t}{e^t}-\frac{\sin t}{e^t}}{-\cos^2 (t)-\sin^2(t)}=\frac{-3\cos t+\sin t}{e^t}$
