The number of distinct real roots of the equation,$\left| \begin{matrix} cosx & sinx & sinx \ sinx & cosx & sinx \ sinx & sinx & cosx \end{matrix} \right| =0$ In t interval $\left[ -\dfrac { \pi }{ 4 } \dfrac { \pi }{ 4 } \right]$ is/are:

Using the determinant formula for a matrix with identical diagonal elements and identical off-diagonal elements, the determinant evaluates to (cosx - sinx)^2(cosx + 2sinx). Setting this equal to zero gives the equations cosx - sinx = 0 and cosx + 2sinx = 0. The first equation simplifies to tanx = 1, which gives the solution x = pi/4 in the given interval. The second equation simplifies to tanx = -1/2, yielding no solutions within the interval [-pi/4, pi/4]. Therefore, there is exactly 1 distinct real root in the specified interval.

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