jmc

The fraction $\frac{1}{(t-1{)}^2+(t+1{)}^2}$ fails to be defined only if the denominator is zero. But $(t-1{)}^2$ and $(t+1{)}^2$ are both nonnegative for all $t$, and are never simultaneously $0$, so their sum is always positive (and, specifically, nonzero). Therefore, the domain of $f(t)$ is all real numbers or, in interval notation, $\boxed{(-\infty ,\infty )}$.