\( 13 \cos 150^{\circ}-\tan 225^{\circ} \sin 270^{\circ} \) \( \sin (60) \cdot \cos 420^{\circ} \)

Let's simplify each expression step by step. ───────────────────────────── Expression 1: 13 cos 150° – tan 225° sin 270° 1. Evaluate cos 150°:   150° = 180° – 30°, so cos 150° = –cos 30° = –(√3/2).   Thus, 13 cos 150° = 13(–√3/2) = –(13√3)/2. 2. Evaluate tan 225°:   225° = 180° + 45°. Since the period of tangent is 180°, tan 225° = tan 45° = 1. 3. Evaluate sin 270°:   sin 270° = –1. 4. Now compute tan 225° sin 270°:   tan 225° sin 270° = 1 × (–1) = –1. 5. Put everything together:   13 cos 150° – tan 225° sin 270° = –(13√3)/2 – (–1) = –(13√3)/2 + 1.   It can also be written as 1 – (13√3)/2. ───────────────────────────── Expression 2: sin 60° · cos 420° 1. Evaluate sin 60°:   sin 60° = √3/2. 2. Evaluate cos 420°:   420° = 360° + 60°, so cos 420° = cos 60° = 1/2. 3. Multiply them:   sin 60° · cos 420° = (√3/2) · (1/2) = √3/4. ───────────────────────────── Final Answers:   • Expression 1 simplifies to: 1 – (13√3)/2   • Expression 2 simplifies to: √3/4