Understanding the Trigonometric Expression

The question asks us to simplify the expression: \( \sin (\alpha + \beta) - 2\sin \alpha \cos \beta + \sin (\alpha - \beta) \).

Applying Trigonometric Identities

To simplify this expression, we need to use the fundamental trigonometric identities for the sine of the sum and difference of two angles. These identities are:

• For the sum of two angles \(\alpha\) and \(\beta\):
  \( \sin (\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \)
• For the difference of two angles \(\alpha\) and \(\beta\):
  \( \sin (\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta \)

Step-by-Step Simplification Process

Let's substitute these identities into the given expression:

Original Expression \( = \sin (\alpha + \beta) - 2\sin \alpha \cos \beta + \sin (\alpha - \beta) \)

Substitute the identities:

Expression \( = (\sin \alpha \cos \beta + \cos \alpha \sin \beta) - 2\sin \alpha \cos \beta + (\sin \alpha \cos \beta - \cos \alpha \sin \beta) \)

Now, let's remove the parentheses and rearrange the terms to group similar components:

Expression \( = \sin \alpha \cos \beta + \cos \alpha \sin \beta - 2\sin \alpha \cos \beta + \sin \alpha \cos \beta - \cos \alpha \sin \beta \)

Group the terms that involve \(\sin \alpha \cos \beta\) and the terms that involve \(\cos \alpha \sin \beta\):

Expression \( = (\sin \alpha \cos \beta + \sin \alpha \cos \beta - 2\sin \alpha \cos \beta) + (\cos \alpha \sin \beta - \cos \alpha \sin \beta) \)

Now, perform the arithmetic within each group:

• The first group is \( \sin \alpha \cos \beta + \sin \alpha \cos \beta - 2\sin \alpha \cos \beta \). Combining the first two terms gives \( 2\sin \alpha \cos \beta \). So the group becomes \( 2\sin \alpha \cos \beta - 2\sin \alpha \cos \beta \), which equals \( 0 \).
• The second group is \( \cos \alpha \sin \beta - \cos \alpha \sin \beta \). This clearly equals \( 0 \).

Substitute these simplified group values back into the expression:

Expression \( = 0 + 0 \)

Expression \( = 0 \)

Therefore, the simplified value of the expression \( \sin (\alpha + \beta) - 2\sin \alpha \cos \beta + \sin (\alpha - \beta) \) is 0.

Revision Table: Key Trigonometric Identities for Sum and Difference

Additional Information: Understanding Trigonometric Simplification

Simplifying trigonometric expressions is a common task in mathematics and physics. It often involves recognizing patterns and applying fundamental identities like the sum and difference formulas, double angle formulas, half angle formulas, and product-to-sum identities. Mastery of these identities is crucial for solving more complex trigonometric problems and equations.

Interestingly, the expression we just simplified, \( \sin (\alpha + \beta) + \sin (\alpha - \beta) - 2\sin \alpha \cos \beta \), can also be seen in relation to the product-to-sum identity: \( \sin(A+B) + \sin(A-B) = 2 \sin A \cos B \). If we rearrange this identity, we get \( \sin(A+B) + \sin(A-B) - 2 \sin A \cos B = 0 \), which directly matches the structure of the problem expression, confirming our result.