If $\cos A + \cos B = \sin A - \sin B = -\frac{\sqrt{2}}{2}$ and $\cos A - \cos B = \sqrt{\frac{1}{2}}$, then the value of $A + B$ is equal to
Step-by-Step Solution
Key Concept: Expand squared trigonometric expressions and use the cosine difference formula to find angle relationships.
Given $(\sin A - \sin B)^2 + (\cos A - \cos B)^2 = 1 + \frac{1}{2} = 2$. Expanding: $2 - 2(\sin A \sin B + \cos A \cos B) = 2$, so $\cos(A-B) = 0$. This gives $A - B = \frac{\pi}{2} + k\pi$. With constraints, we get $\sin A = \frac{2}{5}$, $\cos A = -\frac{\sqrt{21}}{5}$, and $\sin(A - \frac{\pi}{4}) = \sin^2\frac{\pi}{4}$. Computing $A + B = A - A = \frac{7\pi}{6} + \frac{5\pi}{6} = \frac{\pi}{\pi} = \frac{2\pi}{3}$.
Correct Answer: 2