If $\tan A+\sec A=e^x$, find $\cos A$ If $\tan A+\sec A=e^x$, find $\cos A$

Here is what I've tried:

\begin{align}&amp;\frac{\sin A}{\cos A}+\frac{1}{\cos A}&amp;=e^x\\
\implies&amp;\frac{1+\sin A}{\cos A}&amp;=e^x\end{align}

Now, I've squared both sides, but, in vain. How should I continue? Am I proceeding wrong? Any help is much appreciated.
HINT:

As $\sec A+\tan A=e^x$

Now  $\sec A-\tan A=\dfrac1{\sec A+\tan A}=?$

Can you solve for $\sec A?$

Finally $\cos A\cdot\sec A=?$
Given 
tanA+secA=e^x.   ------------(I)
Calculate.   ( secA-tanA) using tan^2A+1=sec^2A
secA-tanA=1/e^x.  ------------(ii)
Add.  (i)&amp;(ii)
2secA=e^x+1/e^x
And then calculate cosA