Answer
The expression in terms of the function $\cos x$ is $\frac{1}{\cos x}$.
Work Step by Step
By using the quotient identity of trigonometry $\tan x=\frac{\sin x}{\cos x}$.
The above expression can be further solved as,
$\frac{\cos x}{1+\sin x}+\tan x=\frac{\cos x}{1+\sin x}+\frac{\sin x}{\cos x}$
Now, simplify the expression on the basis of the common factor. Then we get,
$\begin{align}
& \frac{\cos x}{1+\sin x}+\frac{\sin x}{\cos x}=\frac{{{\cos }^{2}}x+\sin x+{{\sin }^{2}}x}{\left( 1+\sin x \right)\left( \cos x \right)} \\
& =\frac{{{\cos }^{2}}x+{{\sin }^{2}}x+\sin x}{\left( 1+\sin x \right)\left( \cos x \right)}
\end{align}$
Now, apply the Pythagorean identity of trigonometry ${{\sin }^{2}}x+{{\cos }^{2}}x=1$. Then the above expression can be simplified as,
$\begin{align}
& \frac{{{\cos }^{2}}x+{{\sin }^{2}}x+\sin x}{\left( 1+\sin x \right)\left( \cos x \right)}=\frac{1+\sin x}{\left( 1+\sin x \right)\left( \cos x \right)} \\
& =\frac{1}{\cos x}
\end{align}$
Thus, the provided expression is written in the terms of the function $\cos x$ as $\frac{1}{\cos x}$.
Hence, the expression $\frac{\cos x}{1+\sin x}+\tan x$ in terms of the function $\cos x$ is $\frac{1}{\cos x}$.