22. Suppose $A$ and $B$ are two angles such that
$$
\sin A+\sin B=1 \text { and } \cos A+\cos B=0 \text {. }
$$

Find the value of $12 \cos 2 A+4 \cos 2 B$.
22. Ans: 8
$$
\begin{aligned}
\sin A+\sin B & =1 \Longrightarrow \sin A=1-\sin B . \\
\cos A+\cos B & =0 \Longrightarrow \cos A=-\cos B .
\end{aligned}
$$

Thus,
$$
\begin{aligned}
\cos ^{2} A+\sin ^{2} A=1 & \Longrightarrow(-\cos B)^{2}+(1-\sin B)^{2}=1 \\
& \Longrightarrow \cos ^{2} B+1-2 \sin B+\sin ^{2} B=1 \\
& \Longrightarrow \sin B=\frac{1}{2}
\end{aligned}
$$

Thus, $\sin A=1-\frac{1}{2}=\frac{1}{2}$. Therefore,
$$
\begin{aligned}
12 \cos 2 A+4 \cos 2 B & =12\left(1-2 \sin ^{2} A\right)+4\left(1-2 \sin ^{2} B\right) \\
& =12\left(1-2\left(\frac{1}{2}\right)^{2}\right)+4\left(1-2\left(\frac{1}{2}\right)^{2}\right) \\
& =8 .
\end{aligned}
$$