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Sum-to-product identities are especially useful in trigonometry to transform the sum or difference of two trigonometric functions into a product. This can simplify the solving and manipulation of trigonometric expressions. For example:
The sum-to-product identity for sine is:
\[ \sin A + \sin B = 2 \sin \left (\frac{A+B}{2} \right ) \cos \left (\frac{A-B}{2} \right ) \] In our exercise, we're given the sum of two sine functions:
\[ \sin (x+y) + \sin (x-y) \] By directly applying the sum-to-product identities:
Set \[ A = x+y \], and \[ B = x-y \], we then substitute these into our formula, and our expression becomes:
\[ 2 \sin \left (\frac{(x+y)+(x-y)}{2} \right ) \cos \left (\frac{(x+y)-(x-y)}{2} \right ) \] These identities often help in simplifying the trigonometric equations and make verification of identities smoother. They convert sums into more manageable forms, helping us to recognize familiar patterns and relationships.

The sine addition formula is fundamental in trigonometry and helps in expressing the sine of the sum of two angles in terms of their individual sines and cosines. The formula is:
\[ \sin (A + B) = \sin A \cos B + \cos A \sin B \] This formula allows you to break down more complex sine expressions into simpler components. For instance, calculating: \[ \sin (x + y) + \sin (x - y) \] individually using the sine addition formula:
\[ \sin (x + y) = \sin x \cos y + \cos x \sin y \] and
\[ \sin (x - y) = \sin x \cos y - \cos x \sin y \] By summing these results, you get:
\[ (\sin x \cos y + \cos x \sin y) + (\sin x \cos y - \cos x \sin y) = 2 \sin x \cos y \] Thus, verifying that:
\[ \sin (x + y) + \sin (x - y) = 2 \sin x \cos y \] Showing how the sine addition formula can simplify the process of verifying trigonometric identities.

Verification of trigonometric identities is an essential step in understanding and proving the relationships between different trigonometric functions. The objective is to prove that two sides of an equation are equal by using known identities and straightforward algebraic manipulation.
Step-by-step verification of the identity: \[ \sin (x + y) + \sin (x - y) = 2 \sin x \cos y \] 1. **Apply sum-to-product identity:**

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

2. **Simplify arguments of sine and cosine:**
Calculate:
\[ \frac{(x+y)+(x-y)}{2} = x \] and
\[ \frac{(x+y)-(x-y)}{2} = y \] 3. **Conclusion:**
Therefore, the transformed expression simplifies to:
\[ \sin (x+y) + \sin (x-y) = 2 \sin x \cos y \] Each of these steps ensures that the original equation holds true. Verification requires familiarity with various trigonometric identities and the ability to simplify expressions logically.