Answer
See answe below
Work Step by Step
We are given:
$f_1(x)=\sin (x)$
$f_2(x)=\cos x$
$f_3(x)=\tan x$
and $I=(-\frac{\pi}{2}, +\frac{\pi}{2})$
The augmented matrix of this system is:
$W[f_1,f_2,f_3](x)\begin{bmatrix}
\sin x & \cos x & \tan x\\
\cos x& -\sin x & \frac{1}{\cos ^2 x} \\
\sin x & -\cos x & \frac{2 \sin x}{\cos ^3x}
\end{bmatrix} $ for all $x$ in $(-\frac{\pi}{2}, +\frac{\pi}{2})$
Then $W[f_1,f_2,f_3](\frac{\pi}{4})=\begin{bmatrix}
\frac{\sqrt 2}{2}& \frac{\sqrt 2}{2} & 1\\
\frac{\sqrt 2}{2} & -\frac{\sqrt 2}{2}& 2\\
-\frac{\sqrt 2}{2} & -\frac{\sqrt 2}{2} & 4
\end{bmatrix}=\begin{bmatrix}
\frac{\sqrt 2}{2}& -\frac{\sqrt 2}{2} \\
-\frac{\sqrt 2}{2} & -\frac{\sqrt 2}{2}
\end{bmatrix}-2\begin{bmatrix}
\frac{\sqrt 2}{2}& \frac{\sqrt 2}{2} \\
-\frac{\sqrt 2}{2} & -\frac{\sqrt 2}{2}
\end{bmatrix}+4\begin{bmatrix}
\frac{\sqrt 2}{2}& \frac{\sqrt 2}{2} \\
\frac{\sqrt 2}{2} & -\frac{\sqrt 2}{2}
\end{bmatrix}=(-\frac{1}{2}-\frac{1}{2})-2(-\frac{1}{2}+\frac{1}{2})+4(-\frac{1}{2}-\frac{1}{2})=-1-4-0=-5$
Hence, the set of vectors $\{f_1, f_2, f_3 \}$ is linearly independent on $(-\frac{\pi}{2}, +\frac{\pi}{2})$