Question

Suppose $A$ and $B$ are two angles such that $A,Bϵ(0,\pi ),$ and satisfy $\sin A+\sin B=1$ and $\cos A+\cos B=0$ . Then the value of $12\cos 2A+4\cos 2B$ is ________.

Answer 1

$\begin{array}{c}\cos A+\cos B=0\\ \Rightarrow A+B=\pi \\ \sin A+\sin B=1\\ \Rightarrow \sin A=\frac{1}{2}=\sin B\\ 12\left[1-2{\sin }^{2}A\right]+4\left[1-2{\sin }^{2}B\right]\\ =12\left[1-\frac{2}{4}\right]+4\left[1-\frac{2}{4}\right]\\ =6+4\times \frac{1}{2}\\ =8\end{array}$