Real and complex low pass equivalent of band-pass signal
It's not true that any real-valued band pass signal can be written as shown in your first equation:

$$s(t)=A(t)\cos(\omega_c t)\tag{1}$$

The most general representation of a real-valued band pass signal is given by

$$s(t)=\Re\{a(t)e^{j\omega_c t}\}\tag{2}$$

where $a(t)$ is a complex-valued function, called the complex envelop of $s(t)$. It is the complex baseband (low pass) representation of $s(t)$. With $a(t)=a_I(t)+ja_Q(t)$, (2) can be written as

$$s(t)=a_I(t)\cos(\omega_c t)-a_Q(t)\sin(\omega_c t)\tag{3}$$

where $a_I(t)$ and $a_Q(t)$ are called the in-phase and the quadrature components, respectively. Both $a_I(t)$ and $a_Q(t)$ are real-valued baseband (low pass) signals.

A third, equivalent, representation of $s(t)$ is obtained by writing the complex envelop as

$$a(t)=b(t)e^{j\phi(t)}\tag{4}$$

where $b(t)$ is the (real-valued) envelop, and $\phi(t)$ is the phase. Using (4) and (2), the band pass signal $s(t)$ can be written as

$$s(t)=b(t)\cos(\omega_ct+\phi(t))\tag{5}$$

Equations (2), (3) and (5) are three equivalent representations of a general real-valued band pass signal. If you compare representation (5) to Equation (1), you see that Equation (1) is a special case for $\phi(t)=2\pi k$, $k=0,1,\ldots$, i.e. for a constant phase which is zero (or a multiple of $2\pi$). From Equation (3), this is equivalent with the condition that the quadrature component $a_Q(t)$ is zero.

Note that the complex-valued band pass signal

$$\tilde{s}(t)=a(t)e^{j\omega_c t}\tag{6}$$

is the analytic signal mentioned in a comment. It has no negative frequency components and it can be obtained from $s(t)$ by computing

$$\tilde{s}(t)=s(t)+j\mathcal{H}\{s(t)\}\tag{7}$$

where $\mathcal{H}\{\cdot\}$ denotes the Hilbert transform.