The value of the definite integral $\int\limits_{t+2 \pi}^{t+5 \pi / 2}\left\{\sin ^{-1}(\cos x)+\cos ^{-1}(\cos x)\right\} d x$ is equal to

$\frac{\pi^2}{4}$

Since $\sin ^{-1}(\cos x)+\cos ^{-1}(\cos x)$ is a periodic function with period $2 \pi$.

∴  $I=\int\limits_{t+2 \pi}^{t+5 \pi / 2}\left\{\sin ^{-1}(\cos x)+\cos ^{-1}(\cos x)\right\} d x$

$\Rightarrow I=\int\limits_0^{\pi / 2}\left\{\sin ^{-1}(\cos x)+\cos ^{-1}(\cos x)\right\} d x$

$\Rightarrow I=\int\limits_0^{\pi / 2} \frac{\pi}{2} d x=\frac{\pi^2}{4}$