The number of distinct real roots of the folllowing determinant 
The number of distinct real roots of determinant $$
        \begin{vmatrix}
        \csc x & \sec x & \sec x \\
        \sec x & \csc x & \sec x \\
        \sec x & \sec x & \csc x \\
        \end{vmatrix}
=0$$ lies in the interval $\frac{-\pi}{4} \le x \le \frac{\pi}{4}$ is (a) $1$   (b) $2$ (c) $3$ (d) $0$

I tried to solve the determinant and solved it till I got:
 $$(\csc x + 2\sec x)(\csc x - \sec x)^2=0$$
How do I get the final answer?Please offer your assistance, Thank you

We need to solve $\cos(x) + 2\sin(x) = 0$ and $\cos(x) - \sin(x) = 0$. The first equation is solved by setting $u = \cos(x)$ and solving for $u$, i.e.
\begin{align*}
u+2\sqrt{1-u^2} &= 0\\
u &= -2\sqrt{1-u^2} \\
u^2 &= 4(1-u^2) \\
5u^2 &= 4
\end{align*}
and checking which values of $\cos^{-1}(\pm 2/\sqrt{5})$ lie in the specified interval. For the second equation, only $x = \pi/4$ works.
