In radio communication, a message $M(t)$ is not usually transmitted directly. Instead, a technique called Amplitude Modulation $($AM$)$ may be used, in which a carrier wave $C(t)=A_{c}sin({}_{c}t)$ has its amplitude modulated by the message. In short, the signal $S(t)=(1+\frac{M(t)}{A_c})C(t)$ is transmitted instead of $M(t)$ itself.

$($a$)$ Prove for all a and $b$ in $R$ that

$2sin(a)cos(b)=sin(a+b)+sin(a-b).$

$($b$)$ Suppose that you receive an AM radio signal of the form

$S(t)=2sin(98t)+12sin(100t)+2sin(102t).$

Assuming that the message is of the form $M(t)=A_{m}cos({}_{m}t),$ determine expressions for the message and carrier wave that were used to create this signal.