$cosec^{-1}(cos x) $ is defined if

x ∈ [-1, 1] x ∈ R $x = (2n + 1)\frac{\pi}{2},$ n ∈ Z x = n π, n ∈ Z

x = n π, n ∈ Z

We know that $cosec^{-1}x $ is defined for x ∈ (-∞, -1] ∪ [1, ∞). Therefore, $cosec^{-1}(cos x)$ is defined for  $ cos x = ± 1 ⇒$ x = n  π, n ∈ Z.