cosine approximation

Question

For cosine approximation , how to arrive at (9.11) from (9.10) ?

Note: I do not think cosine angle addition formula helps here.

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

Answer 1

The expression in the brackets can be easily written as

$$\left[\frac{1}{M}\left(M\omega_{in} t - K_{VCO}V_m \frac{1}{\omega_{in}}\cos(\omega_{in} t)\right)\right]$$

Now you can use

$$\cos(\omega_{in} t) \approx 1 - \frac{1}{2}\omega^2_{in}t^2$$

And rewrite the brackets.

The rest is trivial algebra.