Does Frequency Modulation uses Envelope Detector?

The modulation involves the concept of variation of the amplitude, frequency or phase components of the carrier signal concerning the instantaneous amplitude of the message signal.
Generally, the carrier signal is a high-frequency component of the sinusoidal.
c(t) = Ac*cos(2πfct)
If the user understands the information without any distortion, then the communication is effective. 
Envelope Detection in AM:
Consider the modulated signal of AM, s(t)=(Ac+m(t))*cos(2πfct). 
Where m(t) is called the message signal.
Here the envelope is called a(t)=(Ac+m(t)).
The component a(t) contains the information of signal m(t) as the magnitude is continuously varying and hence can be detected.

Envelope Detection in FM:
In Frequency Modulation, the standard equation is:
s(t) = Accos(2πfct + 2πKf∫m(t)dt) 
Where Kf is called Frequency Sensitivity.
Considering Single tone modulation i.e., m(t)= Am*cos(2πfmt), Then
s(t) = Accos[2πfct+ (KfAm/fm)sin(2pifmt)] 
Let KfAm/fm = β, Then
s(t) = Accos[2πfct+ βsin(2pifmt)] 
s(t) = Accos(2πfct)*cos(β*sin(2πfmt)) - Acsin(2πfct)*sin(β*sin(2πfmt))

Coming to the detection of the envelope in s(t)   
The inphase component of s(t) is : Ac*cos(β*sin(2πfmt))
The quadrature component of s(t) is : Ac*sin(β*sin(2πfmt))

Then the magnitude of the envelope of s(t) is : 

which in turn results in |Ac|.
Generally, the detector captures the message signal if there is variation in the envelope. But in the case of Frequency modulated signal, the envelope remains constant i.e; |Ac|. So the information of message signal cannot be extracted and hence remains clueless.
There remains a path for demodulation. A discriminator is used in this case and hence the message signal can be detected.