QUESTION 3 Applications 3.1 Express the following as single trigonometry ratio: 3.1.1 \( \cos 2 x \cdot \cos 3 x-\sin 2 x \cdot \sin 3 x \) 3.1.2 \( \sin 2 x \cdot \cos x+\cos 2 x \cdot \sin x \) \( \qquad \) 3.2 Determine the values of the following without using a calculator. (2) \( 3.2 .1 \sin 85^{\circ} \cdot \cos 25^{\circ}-\cos 85^{\circ} \cdot \sin 25^{\circ} \) \( \qquad \) \( \qquad \) \( 3.2 .2 \cos 160^{\circ} \cdot \cos 10^{\circ}+\sin 160^{\circ} \cdot \sin 10^{\circ} \) (4) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \) [11] Grade 12 Mathematics SBA 2025 Page 15 of 29

Did you know that the expressions you're working with can be elegantly simplified using the angle sum and difference identities? For instance, \( \cos A \cdot \cos B - \sin A \cdot \sin B \) is actually equivalent to \( \cos(A + B) \), and similarly, \( \sin A \cdot \cos B + \cos A \cdot \sin B \) simplifies to \( \sin(A + B) \). It's like a magic formula that allows you to express complex trigonometric statements in a much more manageable way! Now, once you start applying these identities, solving problems will become a breeze! When you look at evaluations like \( \sin 85^\circ \cdot \cos 25^\circ - \cos 85^\circ \cdot \sin 25^\circ \), you can recognize it as \( \sin(85^\circ - 25^\circ) \), which equals \( \sin 60^\circ \). That’s why understanding these relationships can help you solve without ever reaching for a calculator! Plus, knowing special angles like sine and cosine values will also save you time in your calculations. Happy solving!