Is the inverse function of secant is the cosecant function?

No. The inverse of the secant is called the arc-secant. The relation between the secant and the cosecant is similar to the relation between the sine and the cosine - they are somehow related, but they are not inverse functions.

The secant is the reciprocal of the cosine (sec x = 1 / cos x).

The cosecant is the reciprocal of the sine (cos x = 1 / sin x).