These are the identity-level properties (Properties I–V of the chapter) that let an inverse-trig expression be simplified WITHOUT drawing a triangle or invoking a sum formula — they hold strictly within the principal value branches.
Property I — undoing the outer inverse. f−1(f(θ))=θ holds only when θ already lies in f's principal domain: sin−1(sinθ)=θ if θ∈[−2π,2π]; cos−1(cosθ)=θ if θ∈[0,π]; tan−1(tanθ)=θ if θ∈(−2π,2π); similarly for cosec−1,sec−1,cot−1 on their own principal domains. If θ is OUTSIDE the principal domain, f−1(f(θ))=θ — instead, use periodicity/symmetry to rewrite f(θ) as f(θ1) for some θ1 that IS inside the principal domain, then f−1(f(θ))=θ1. E.g. sin−1(sin65π)=sin−1(sin(π−6π))=sin−1(sin6π)=6π, since 6π∈[−2π,2π].
Property II — undoing the inner inverse. f(f−1(x))=x holds throughout f−1's entire domain with no extra condition: sin(sin−1x)=x for x∈[−1,1]; cos(cos−1x)=x for x∈[−1,1]; tan(tan−1x)=x for every real x; and likewise for the other three on their own domains. This is the direct definition of "inverse" and never needs a range check.
Property III (reciprocal identities). sin−1(x1)=cosec−1x and cos−1(x1)=sec−1x, both for x∈R∖(−1,1); and tan−1(x1)=cot−1x if x>0, but =−π+cot−1x if x<0 (the sign correction is needed here because tan−1(x1) stays in (−2π,0) for x<0 while cot−1x lands in (2π,π) — different branches of the same underlying angle).
Property IV (reflection identities — negating the argument). sin−1(−x)=−sin−1x; tan−1(−x)=−tan−1x; cosec−1(−x)=−cosec−1x (all three odd); but cos−1(−x)=π−cos−1x; sec−1(−x)=π−sec−1x; cot−1(−x)=π−cot−1x (all three pick up a π−, since their principal range [0,π]-type interval isn't symmetric about 0). …