Suppose that $\sin a + \sin b = \sqrt{\frac{5}{3}}$ and $\cos a + \cos b = 1$. What is $\cos (a - b)$? $\mathrm{(A)}\ \sqrt{\frac{5}{3}} - 1\qquad$ $ \mathrm{(B)}\ \frac 13\qquad$ $ \mathrm{(C)}\ \frac 12\qquad$ $ \mathrm{(D)}\ \frac 23\qquad$ $ \mathrm{(E)}\ 1$
Solution
Squaring both equations and adding: $(\sin a+\sin b)^2+(\cos a+\cos b)^2=2+2\cos(a-b)$.
This equals $\frac53+1=\frac83$, so $2\cos(a-b)=\frac83-2=\frac23$.
