QUESTION 3 Applications 3.1 Express the following as single trigonometry ratio: 3.1.1 \( \cos 2 x \cdot \cos 3 x-\sin 2 x \cdot \sin 3 x \) \[ \cos (2 x+3 x)=\cos (5 x) \] 3.1 .2 \[ \begin{array}{l} \sin 2 x \cdot \cos x+\cos 2 x \cdot \sin x \\ \sin (2 x+x)=\sin 3 x \end{array} \] 3.2 Determine the values of the following without using a calculator. 3.2.1 \( \sin 85^{\circ} \cdot \cos 25^{\circ}-\cos 85^{\circ} \cdot \sin 25^{\circ} \) \[ \begin{array}{l} \sin 85^{\circ} \cdot \cos 25^{\circ}-\cos 85^{\circ} \cdot \sin 25^{\circ} \\ \sin A \cdot \cos B-\cos A \sin B=\sin (A-B) \\ \sin 85^{\circ} \cdot \cos 25-\cos 85^{\circ} \sin 25=\sin \left(85^{\circ}-25^{\circ}\right)=\sin \\ \therefore \sin 60=\frac{\frac{\pi}{3}}{2} \end{array} \] 3.2.2 \( \cos 160^{\circ} \cdot \cos 10^{\circ}+\sin 160^{\circ} \cdot \sin 10^{\circ} \) (4) \( \qquad \) \( \qquad \) \( \qquad \) \( \qquad \)

Below are the step‐by‐step solutions: 3.1 Express each expression as a single trigonometric function. 3.1.1  We use the cosine addition formula:   cos A cos B – sin A sin B = cos (A + B)  Let A = 2x and B = 3x. Then,   cos 2x · cos 3x – sin 2x · sin 3x = cos (2x + 3x) = cos 5x. 3.1.2  We use the sine addition formula:   sin A cos B + cos A sin B = sin (A + B)  Let A = 2x and B = x. Then,   sin 2x · cos x + cos 2x · sin x = sin (2x + x) = sin 3x. 3.2 Compute each value without using a calculator. 3.2.1  We recognize the formula:   sin A cos B – cos A sin B = sin (A – B)  Let A = 85° and B = 25°. Then,   sin 85° · cos 25° – cos 85° · sin 25° = sin (85° – 25°) = sin 60°.  Recall that sin 60° = (√3)/2. 3.2.2  We use the cosine identity:   cos A cos B + sin A sin B = cos (A – B)  Let A = 160° and B = 10°. Then,   cos 160° · cos 10° + sin 160° · sin 10° = cos (160° – 10°) = cos 150°.  Using the identity cos (180° – θ) = –cos θ, with θ = 30°,   cos 150° = –cos 30° = –(√3)/2. Summary of answers:  3.1.1 cos 5x  3.1.2 sin 3x  3.2.1 (√3)/2  3.2.2 –(√3)/2