Verify that each equation is an identity.
sin(x + y) + sin(x - y) = 2 sin x cos y

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Recall the sum-to-product identities for sine functions. Specifically, the identity for the sum of two sine terms is: \(\sin A + \sin B = 2 \sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right)\).

Identify \(A\) and \(B\) in the given equation. Here, \(A = x + y\) and \(B = x - y\).

Apply the sum-to-product identity to the left-hand side (LHS): \(\sin(x + y) + \sin(x - y) = 2 \sin \left( \frac{(x + y) + (x - y)}{2} \right) \cos \left( \frac{(x + y) - (x - y)}{2} \right)\).

Simplify the arguments inside the sine and cosine functions: \(\frac{(x + y) + (x - y)}{2} = \frac{2x}{2} = x\) and \(\frac{(x + y) - (x - y)}{2} = \frac{2y}{2} = y\).

Rewrite the expression after simplification: \(2 \sin x \cos y\), which matches the right-hand side (RHS) of the original equation, thus verifying the identity.

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Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. They are used to simplify expressions or prove equivalences, such as verifying that two sides of an equation represent the same function.

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The sum and difference formulas express sine of sums or differences of angles: sin(a ± b) = sin a cos b ± cos a sin b. These formulas allow breaking down complex expressions into simpler parts, essential for verifying identities like sin(x + y) + sin(x - y).

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After applying sum and difference formulas, combining like terms (such as sin x cos y terms) simplifies the expression. Recognizing and grouping these terms correctly is crucial to show that both sides of the identity are equal.

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