To solve the problem, we need to find the derivative of the given function $y = \frac{\pi}{5} \sin(\theta) - \sin(\theta)$ with respect to $\theta$.

Step-by-Step Solution:

1. Differentiate each term separately:

   • The derivative of $\frac{\pi}{5} \sin(\theta)$ with respect to $\theta$: $\frac{d}{d\theta} \left( \frac{\pi}{5} \sin(\theta) \right) = \frac{\pi}{5} \cos(\theta)$
   • The derivative of $-\sin(\theta)$ with respect to $\theta$: $\frac{d}{d\theta} (-\sin(\theta)) = -\cos(\theta)$

2. Combine the results: $y' = \frac{\pi}{5} \cos(\theta) - \cos(\theta)$

3. Factor out the common term $\cos(\theta)$: $y' = \cos(\theta) \left(\frac{\pi}{5} - 1\right)$

Final Answer:

$y' = \left(\frac{\pi}{5} - 1\right) \cos(\theta)$

Would you like any further details on this solution? Here are some related questions:

1. What is the derivative of $\sin(\theta)$ with respect to $\theta$?
2. How do you factor common terms in a mathematical expression?
3. What is the significance of the cosine function in trigonometric derivatives?
4. How does the constant factor affect the differentiation process?
5. Can you explain the chain rule in differentiation?

Tip: When differentiating trigonometric functions, remember that the derivative of $\sin(\theta)$ is $\cos(\theta)$ and the derivative of $\cos(\theta)$ is $-\sin(\theta)$.