Chain Rule with 1/cos(x)

Question

In the statement of the attached snapshot it states:

$1/\cos(x)$ is made up of $1/g$ and $\cos()$:

$$f(g) = 1/g$$

How is $1/\cos(x)$ partially made up of $1/g$?

Where does $1/g$ come from?

Answer 1

Let $g(x) = \cos x$. Then $\frac{1}{\cos x} = \frac{1}{g}$. Then $$\left(\frac{1}{\cos x}\right)' = \left(\frac{1}{g}\right)' = -\frac{1}{g^2}g' = -\frac{1}{\cos ^2x}(-\sin x) = \frac{\sin x}{\cos^2x}$$