(1)求 $\left(1+\tan ^{2} 15^{\circ}\right) \cos ^{2} 15^{\circ}$ 的值;

(2)已知 $\theta$ 是第三象限角，且 $\sin ^{4} \theta+\cos ^{4} \theta=\frac{5}{9}$, 求 $\sin \theta \cos \theta$ 的值.
(1) $\left(1+\tan ^{2} 15^{\circ}\right) \cos ^{2} 15^{\circ}=\left(1+\frac{\sin ^{2} 15^{\circ}}{\cos ^{2} 15^{\circ}}\right) \cos ^{2} 15^{\circ}=\cos ^{2} 15^{\circ}+\sin ^{2} 15^{\circ}=1$.

(2) $\sin ^{4} \vartheta+\cos ^{4} \vartheta=\left(\sin ^{2} \vartheta+\cos ^{2} \vartheta\right)^{2}-2 \sin ^{2} \vartheta \cos ^{2} \vartheta=1-2 \sin ^{2} \vartheta \cos ^{2} \vartheta==^{\frac{5}{9}}$,

$\therefore \sin ^{2} \vartheta \cos ^{2} \vartheta={ }^{\frac{2}{9}},: \vartheta$ 是第三象限角, $\therefore \sin \vartheta \cos \vartheta=\frac{\sqrt{2}}{3}$.