Answer
See the proof below.
Work Step by Step
\begin{align*}
\frac{d }{dx} \sec x&=\frac{d }{dx} \frac{1}{\cos x}\\
&=\frac{\cos x(0)+\sin x}{\cos^2 x}\\
&=\frac{\sin x}{\cos^2 x}\\
&=\frac{1}{\cos x}\frac{\sin x}{\cos x}\\
&=\sec x \tan x.
\end{align*}
