7.11 Let $x=\cos 36^{\circ}-\cos 72^{\circ}$. Then $x$ equals
(A) $\frac{1}{3}$.
(B) $\frac{1}{2}$.
(C) $3-\sqrt{6}$.
(D) $2 \sqrt{3}-3$.
(E) None of these.
(26th American High School Mathematics Examination, 1975)
[Solution] Let $u=\cos 36^{\circ}$, and $v=\cos 72^{\circ}$.
From $\cos 2 \theta=2 \cos ^{2} \theta-1$ and $\cos 2 \theta=1-2 \sin ^{2} \theta$, substituting $\theta=36^{\circ}$ into the first equation, and $\theta=18^{\circ}$ into the second equation, we get
$v=2 u^{2}-1$ and $u=1-2 v^{2}$.
Adding the two equations, we have
$$
u+v=2\left(u^{2}-v^{2}\right),
$$

which simplifies to
$$
u-v=\frac{1}{2} \text {. }
$$

Therefore, the answer is $(B)$.