Express $\sin (a + b) - \sin (a - b)$ as the product of trigonometric functions.
By sum-to-product,
\[\sin (a + b) - \sin (a - b) = \boxed{2 \sin b \cos a}.\]
2\sinb\cosa
Level 2
Precalculus
Problem: Express $\sin (a + b) - \sin (a - b)$ as the product of trigonometric functions.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To express $\sin (a + b) - \sin (a - b)$ as a product, we can utilize the sine subtraction and addition formulas. The sine addition formula states that $\sin(a + b) = \sin a \cos b + \cos a \sin b$, and the sine subtraction formula states that $\sin(a - b) = \sin a \cos b - \cos a \sin b$. 

First, we apply these formulas to the expression:

1. Substitute the formulas into the expression:
   $\sin(a + b) - \sin(a - b) = (\sin a \cos b + \cos a \sin b) - (\sin a \cos b - \cos a \sin b)$.

2. Simplify the expression:
   The $\sin a \cos b$ terms cancel out, leaving us with:
   $\sin(a + b) - \sin(a - b) = \cos a \sin b + \cos a \sin b = 2 \cos a \sin b$.

3. Rearranging the terms gives us the final result:
   $2 \sin b \cos a$.

Thus, the expression $\sin (a + b) - \sin (a - b)$ can be expressed as the product of trigonometric functions, specifically $2 \sin b \cos a$.<|END_OF_LATENT|> The final answer is $2\sinb\cosa$.