Find $\sin A$ and $\cos A$ if $\tan A+\sec A=4 $ How to find $\sin A$ and $\cos A$ if
$$\tan A+\sec A=4 ?$$
I tried to find it by $\tan A=\dfrac{\sin A}{\cos A}$ and $\sec A=\dfrac{1}{\cos A}$, therefore
$$\tan A+\sec A=\frac{\sin A+1}{\cos A}=4,$$
which implies 
$$\sin A+1=4\cos A.$$
Then what to do?

Together with
$$\sin A+1=4\cos A $$
you can use $\sin^2A+\cos^2A=1$ as
$$(\sin A+1)(\sin A-1)=\cos^2 A\ .$$
Putting the two together you easily get
$$4\cos A(\sin A-1)=\cos^2 A\ ,$$
and hence
$$4\sin A-4=\cos A\ .$$
You now  just have to solve the linear system in $\sin A$ and $\cos A$:
$$\begin{cases}
4\sin A-4=\cos A\\
\sin A+1=4\cos A
\end{cases}$$
