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

Verified step by step guidance

1. Start by using the sum-to-product identities for sine: \( \sin(A) + \sin(B) = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \).
2. Set \( A = x + y \) and \( B = x - y \).
3. Calculate \( \frac{A + B}{2} = \frac{(x + y) + (x - y)}{2} = x \).
4. Calculate \( \frac{A - B}{2} = \frac{(x + y) - (x - y)}{2} = y \).
5. Substitute back into the identity: \( 2 \sin(x) \cos(y) \), which matches the right side of the given equation, verifying the identity.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variables involved. They are fundamental in simplifying expressions and solving equations in trigonometry. Common identities include the Pythagorean identities, angle sum and difference identities, and double angle identities, which provide relationships between the sine, cosine, and tangent functions.

Angle Sum and Difference Formulas

The angle sum and difference formulas express the sine and cosine of the sum or difference of two angles in terms of the sines and cosines of the individual angles. For example, sin(x + y) = sin x cos y + cos x sin y and sin(x - y) = sin x cos y - cos x sin y. These formulas are essential for verifying identities involving sums and differences of angles.

Verification of Identities

Verifying trigonometric identities involves manipulating one side of the equation to show that it is equivalent to the other side. This process often requires the use of known identities, algebraic techniques, and sometimes factoring or expanding expressions. The goal is to demonstrate that both sides represent the same value for all permissible values of the variables.