1. **Problem:** Prove that $$\frac{\sin(x-y)}{\cos x \cos y} + \frac{\sin(y-z)}{\cos y \cos z} + \frac{\sin(z-x)}{\cos z \cos x} = 0$$ 2. **Formula and rules:** Use the sine subtraction formula: $$\sin(a-b) = \sin a \cos b - \cos a \sin b$$ 3. **Step-by-step proof:** - Write each term using the sine subtraction formula: $$\frac{\sin x \cos y - \cos x \sin y}{\cos x \cos y} + \frac{\sin y \cos z - \cos y \sin z}{\cos y \cos z} + \frac{\sin z \cos x - \cos z \sin x}{\cos z \cos x}$$ - Split each fraction: $$\frac{\sin x \cos y}{\cos x \cos y} - \frac{\cos x \sin y}{\cos x \cos y} + \frac{\sin y \cos z}{\cos y \cos z} - \frac{\cos y \sin z}{\cos y \cos z} + \frac{\sin z \cos x}{\cos z \cos x} - \frac{\cos z \sin x}{\cos z \cos x}$$ - Simplify each term: $$\frac{\sin x}{\cos x} - \frac{\sin y}{\cos y} + \frac{\sin y}{\cos y} - \frac{\sin z}{\cos z} + \frac{\sin z}{\cos z} - \frac{\sin x}{\cos x}$$ - Notice terms cancel: $$\left(\frac{\sin x}{\cos x} - \frac{\sin x}{\cos x}\right) + \left(- \frac{\sin y}{\cos y} + \frac{\sin y}{\cos y}\right) + \left(- \frac{\sin z}{\cos z} + \frac{\sin z}{\cos z}\right) = 0$$ 4. **Conclusion:** The expression equals zero, as required. --- 1. **Problem:** Draw graph of $$y = \cos x$$ for $$0 \leq x \leq \pi$$. 2. **Explanation:** The cosine function starts at 1 when $$x=0$$ and decreases to -1 at $$x=\pi$$. 3. **Graph features:** The curve is smooth, continuous, and decreases monotonically from 1 to -1 over the interval. --- 1. **Problem:** Find equation of line passing through points (1, 2) and (-3, 1). 2. **Formula:** Slope $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ and line equation $$y - y_1 = m(x - x_1)$$. 3. **Calculate slope:** $$m = \frac{1 - 2}{-3 - 1} = \frac{-1}{-4} = \frac{1}{4}$$ 4. **Equation:** $$y - 2 = \frac{1}{4}(x - 1)$$ 5. **Simplify:** $$y = \frac{1}{4}x - \frac{1}{4} + 2 = \frac{1}{4}x + \frac{7}{4}$$ **Final answer:** $$y = \frac{1}{4}x + \frac{7}{4}$$