Problem 31 - Algebra and Trigonometry

Question

\(23-40\) . Prove the identity. $$ \sin (x+y)-\sin (x-y)=2 \cos x \sin y $$

Step-by-Step Solution

Answer

The identity holds: \( \sin(x+y) - \sin(x-y) = 2 \cos x \sin y \).

Step 1: Apply Sine Addition Formula

We know the sine addition formula is given by \( \sin(a + b) = \sin a \cos b + \cos a \sin b \). Begin by applying it to \( \sin(x + y) \).

Step 2: Apply Sine Subtraction Formula

Similarly, for \( \sin(x - y) \), use the sine subtraction formula \( \sin(a - b) = \sin a \cos b - \cos a \sin b \).

Step 3: Express \( \sin(x+y) - \sin(x-y) \) Using Formulas

Substitute both formulas from Steps 1 and 2: \( \sin(x+y) - \sin(x-y) = (\sin x \cos y + \cos x \sin y) - (\sin x \cos y - \cos x \sin y) \).

Step 4: Simplify the Expression

Notice that \( \sin x \cos y \) terms cancel each other out. Simplify to: \( \sin(x+y) - \sin(x-y) = 2 \cos x \sin y \).

Key Concepts

Sine Addition Formula

The sine addition formula is a key trigonometric identity that helps you find the sine of the sum of two angles. In simpler terms, if you want to calculate \( \sin(a + b) \), this formula breaks it down into a combination of sines and cosines of the individual angles. The formula is: \[ \sin(a + b) = \sin a \cos b + \cos a \sin b \] This makes the formula incredibly useful as it allows us to work with more manageable components.

• \( \sin a \) represents the sine of the first angle, \( a \).
• \( \cos b \) represents the cosine of the second angle, \( b \).
• \( \cos a \sin b \) acts as a balancing component by mixing the other trigonometric functions.

Understanding how to apply this formula is essential for simplifying complex trigonometric expressions, particularly when the problem involves angle sums like in the given exercise.

Sine Subtraction Formula

The sine subtraction formula is closely related to the addition formula but is focused on calculating the sine of the difference between two angles. The formula is expressed as: \[ \sin(a - b) = \sin a \cos b - \cos a \sin b \] This formula mirrors the addition formula except for the sign in the middle, which is a subtraction instead of addition. It helps to understand the following:

• When subtracting angles, the cosine component that couples with the sine switches its sign, leading to a different result than the addition.
• The terms will still follow the sine-cosine and cosine-sine pattern as in the addition formula.

Applying this formula allows you to simplify expressions like \( \sin(x - y) \), which is one half of our original exercise. By breaking it into simpler trigonometric components, you facilitate further simplification and proofs involving these identities.

Trigonometric Simplification

Trigonometric simplification involves reducing complex trigonometric expressions into simpler or more recognizable forms. In many problems, like the one you've encountered, you need to use well-known identities such as sine addition and subtraction formulas.

• First, apply the relevant identities to replace the original complex expression with basic trigonometric functions.
• Then, look for chances to combine like terms or cancel them out.

In the original exercise, after applying the sine addition and subtraction formulas, we have: \[ \sin(x + y) - \sin(x - y) = (\sin x \cos y + \cos x \sin y) - (\sin x \cos y - \cos x \sin y) \] You can see that the \( \sin x \cos y \) terms cancel each other out naturally, leaving the simplified form: \[ 2 \cos x \sin y \] This result provides a cleaner and more understandable expression. Simplification helps not just in finding solutions efficiently but also in verifying the correctness of your approach.