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Express the following as a single trigonometric ratio:

(1) $\sin 2 \theta \cdot \cos \theta+\cos 2 \theta \cdot \sin \theta$

(2) $\cos 70^{\circ} \cdot \cos x+\sin 70^{\circ} \cdot \sin x$

(3) $\cos x \cdot \sin 3 x-\cos 3 x \cdot \sin x$

(4) $\sin 3 \theta \cdot \sin 2 \theta-\cos 3 \theta \cdot \cos 2 \theta$
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(a) (1) $\sin 2 \theta \cdot \cos \theta+\cos 2 \theta \cdot \sin \theta=\sin (2 \theta+\theta)=\sin 3 \theta$
(2) $\cos 70^{\circ} \cdot \cos x+\sin 70^{\circ} \cdot \sin x=\cos \left(70^{\circ}-x\right)$
(3)
$\begin{array}{ll} \cos x \cdot \sin 3 x-\cos 3 x \cdot \sin x & \\ =\sin 3 x \cdot \cos x-\cos 3 x \cdot \sin x & \text { (rewrite } \cos x \cdot \sin 3 x \text { as } \sin 3 x \cdot \cos x) \\ =\sin (3 x-x) & (\sin \mathrm{A} \cdot \cos \mathrm{B}-\cos \mathrm{A} \cdot \sin \mathrm{B}=\sin (\mathrm{A}-\mathrm{B})) \\ =\sin 2 x & \end{array}$

(4)
\begin{aligned} & \sin 3 \theta \cdot \sin 2 \theta-\cos 3 \theta \cdot \cos 2 \theta \\ & =-(\cos 3 \theta \cdot \cos 2 \theta-\sin 3 \theta \cdot \sin 2 \theta) \\ & =-\cos (3 \theta+2 \theta)=-\cos 5 \theta \quad \quad(\cos A \cdot \cos B-\sin A \cdot \sin B=\cos (A+B)) \end{aligned}
by Diamond (71,587 points)

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