Evaluate each of the following:

As we know cosec(–θ) = –cosecθ

∴ cosec = –cosec

–cosec can be written as –cosec

Also,

cosec(2π+θ) = cosecθ

∴ –cosec = –cosec

As we know –cosec(θ) = cosec(–θ)

∴ –cosec = cosec

Now the question becomes cosec^–1(cosec)

cosec^–1(cosec x) = x

Provided x ϵ –{0}

∴ we can write cosec^–1(cosec.