1. **Stating the problem:** We want to verify if the formula for the sum of sines $$\sum_{k=1}^n \sin(kx) = \frac{\sin\left(\frac{nx}{2}\right) \cdot \sin\left(\frac{(n+1)x}{2}\right)}{\sin\left(\frac{x}{2}\right)}$$ is mathematically correct and can be used to solve problems. 2. **Recall the known formula:** The sum of sines of an arithmetic sequence is given by $$\sum_{k=1}^n \sin(kx) = \frac{\sin\left(\frac{nx}{2}\right) \sin\left(\frac{(n+1)x}{2}\right)}{\sin\left(\frac{x}{2}\right)}$$ This is a standard trigonometric identity derived from the formula for the sum of a geometric series applied to complex exponentials. 3. **Explanation:** - The numerator contains the product of two sine terms involving $n$ and $n+1$. - The denominator is $\sin\left(\frac{x}{2}\right)$, which normalizes the sum. 4. **Verification by example:** For $n=1$, the sum is $\sin(x)$. Plugging into the formula: $$\frac{\sin\left(\frac{1 \cdot x}{2}\right) \sin\left(\frac{2x}{2}\right)}{\sin\left(\frac{x}{2}\right)} = \frac{\sin\left(\frac{x}{2}\right) \sin(x)}{\sin\left(\frac{x}{2}\right)} = \sin(x)$$ which matches the sum. 5. **Conclusion:** The formula is mathematically correct and can be used to solve problems involving sums of sine functions with linearly increasing arguments. **Final answer:** $$\sum_{k=1}^n \sin(kx) = \frac{\sin\left(\frac{nx}{2}\right) \sin\left(\frac{(n+1)x}{2}\right)}{\sin\left(\frac{x}{2}\right)}$$