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?

Rearranging the identity $1 + \tan^2A = \sec^2A$ yields
$$\sec^2A - \tan^2A = 1$$
Factoring yields 
$$(\sec A + \tan A)(\sec A - \tan A) = 1$$
Since $\sec A + \tan A = 4$, we have 
$$4(\sec A - \tan A) = 1$$
which yields the system of equations 
\begin{align*}
\sec A + \tan A & = 4\\
\sec A - \tan A & = \frac{1}{4}
\end{align*}
Solve the system for $\sec A$ and $\tan A$, then use the identities 
\begin{align*}
\cos A & = \frac{1}{\sec A}\\
\sin A & = \tan A\cos A
\end{align*}
to solve for sine and cosine.
