求函数 $f(t)=\left\{\begin{array}{ll}\cos t, & |t| < \frac{\pi}{2} \\ 0, & |t|>\frac{\pi}{2}\end{array}\right.$ 的傅氏变换．并证明广义积分
$$
\int_0^{+\infty} \frac{\cos \frac{\omega \pi}{2} \cos \omega t}{1-\omega^2} \mathrm{~d} \omega=\frac{\pi}{2} \cos t, \quad|t| < \frac{\pi}{2}
$$