We know that $T_{r}\left ( A \right )$ $=$ $Trace\left ( A \right )$ $=$ sum of principle diagonal elements $=$ sum of eigen values of the matrix.
$\therefore$ $2$$cost$ $+$ $1$ $=$ $\sqrt{2}$ $+$ $1$
$\Rightarrow$ $cost$ $=$ $\frac{1}{\sqrt{2}}$
$\Rightarrow$ $t$ $=$ $-$ $\frac{\pi }{4}$ or $\frac{\pi }{4}$ (for the interval $\left [ -\pi ,\pi \right ]$ , $cosx$ is positive between $\left [ -\frac{\pi }{2},\frac{\pi}{2} \right ]$).
$\therefore$ Option C is the correct answer.