  Let $q$ be a power of a prime and $\mathbb{F}_q$ be a finite field with $q$
elements. In this paper, we propose four families of infinite classes of
permutation trinomials having the form $cx-x^s + x^{qs}$ over
$\mathbb{F}_{q^2}$, and investigate the relationship between this type of
permutation polynomials with that of the form $(x^q-x+\delta)^s+cx$. Based on
this relation, many classes of permutation trinomials having the form
$(x^q-x+\delta)^s+cx$ without restriction on $\delta$ over $\mathbb{F}_{q^2}$
are derived from known permutation trinomials having the form $cx-x^s +
x^{qs}$.
