To transform the product $\sin(x) \sin(3x)$ into a sum, we can use the product-to-sum identity for sines:

$\sin A \sin B = \frac{1}{2} [\cos(A - B) - \cos(A + B)]$

In this case, let $A = x$ and $B = 3x$. Applying the identity:

$\sin(x) \sin(3x) = \frac{1}{2} \left[ \cos(x - 3x) - \cos(x + 3x) \right]$

Simplifying the arguments of the cosine terms:

$= \frac{1}{2} \left[ \cos(-2x) - \cos(4x) \right]$

Since $\cos(-2x) = \cos(2x)$ (because cosine is an even function), we have:

$\sin(x) \sin(3x) = \frac{1}{2} \left[ \cos(2x) - \cos(4x) \right]$

So, the expression for $\sin(x) \sin(3x)$ as a sum is:

$\sin(x) \sin(3x) = \frac{1}{2} \left[ \cos(2x) - \cos(4x) \right]$

Would you like further details on this transformation or have any questions? Here are some related questions for you:

1. How does the product-to-sum identity apply to other trigonometric functions like cosine and tangent?
2. What are some other examples of using product-to-sum identities in trigonometry?
3. Can you derive a product-to-sum identity for cosine and sine?
4. How would you simplify expressions like $\sin(x) \cos(x)$ using identities?
5. How can product-to-sum identities be used in Fourier series analysis?

Tip: Practice recognizing patterns in trigonometric identities to help with simplifying complex expressions.