Answer
$$A= \frac{2\pi}{3} +2 n\pi,\ \ \text{or}\ \ \frac{7\pi}{6}+2 n\pi$$
where $n $ is integer
Work Step by Step
Given
$$ \cos \left(A+\frac{\pi}{12}\right) =\left( \frac{-\sqrt{2}}{2}\right)$$
Since
\begin{align*}
A+\frac{\pi}{12}&=\cos^{-1}\left( \frac{-\sqrt{2}}{2}\right)\\
A+\frac{\pi}{12}&=\frac{3 \pi}{4}\ \ \text{or}\ \ \frac{5\pi}{4}\\
A&=\frac{2\pi}{3},\ \ \text{or}\ \ \frac{7\pi}{6}
\end{align*}
Since the period of $\cos$ function is $2\pi $ , then the general solution is
$$A= \frac{2\pi}{3} +2 n\pi,\ \ \text{or}\ \ \frac{7\pi}{6}+2 n\pi$$
where $n $ is integer