In Exercises 45–46, express each sum or difference as a product. If possible, find this product's exact value. sin 2x - sin 4x

Verified step by step guidance

1. Recall the sine difference identity for expressing the difference of sines as a product: \(\sin A - \sin B = 2 \cos \left( \frac{A + B}{2} \right) \sin \left( \frac{A - B}{2} \right)\).
2. Identify \(A\) and \(B\) in the expression \(\sin 2x - \sin 4x\) as \(A = 2x\) and \(B = 4x\).
3. Apply the formula by substituting \(A\) and \(B\): \(\sin 2x - \sin 4x = 2 \cos \left( \frac{2x + 4x}{2} \right) \sin \left( \frac{2x - 4x}{2} \right)\).
4. Simplify the arguments inside the cosine and sine functions: \(\cos \left( 3x \right)\) and \(\sin \left( -x \right)\).
5. Use the odd property of sine, \(\sin(-x) = -\sin x\), to rewrite the expression as a product involving \(\cos 3x\) and \(\sin x\).

Key Concepts

Sum-to-Product Identities

Sum-to-product identities transform sums or differences of sine and cosine functions into products. For sine differences, the identity is sin A - sin B = 2 cos((A+B)/2) sin((A-B)/2). This simplifies expressions and helps in solving or evaluating trigonometric problems.

Angle Substitution and Simplification

After applying identities, substituting the given angles correctly is crucial. Simplifying expressions like (2x + 4x)/2 and (2x - 4x)/2 ensures accurate transformation. This step is essential for reducing the expression to a product form.

Exact Values of Trigonometric Functions

Finding the exact value of a trigonometric expression involves knowing standard angle values and their sine or cosine results. For example, angles like 0°, 30°, 45°, 60°, and 90° have well-known exact sine and cosine values, which help evaluate the product precisely.