For the function f(x)=sin x, show with the aid of elementary formula sin² A=½(1-cos 2A) that f(x+y)-f(x)= cos x sin y-2 sin x sin² (½y)

To show that \( f(x+y) - f(x) = \cos x \sin y - 2 \sin x \sin^2 \left( \frac{y}{2} \right) \) using the identity \( \sin^2 A = \frac{1 - \cos 2A}{2} \), let's proceed step by step.

### Step 1: Express \( f(x+y) \) using the sum of angles formula

Given the function \( f(x) = \sin x \), we can write:

\[
f(x+y) = \sin(x + y) = \sin x \cos y + \cos x \sin y
\]

### Step 2: Calculate \( f(x+y) - f(x) \)

Now, subtract \( f(x) \) from \( f(x+y) \):

\[
f(x+y) - f(x) = \sin(x + y) - \sin x
\]
\[
= (\sin x \cos y + \cos x \sin y) - \sin x
\]

This can be rearranged as:

\[
f(x+y) - f(x) = \sin x \cos y - \sin x + \cos x \sin y
\]

Factor \( \sin x \) from the first two terms:

\[
f(x+y) - f(x) = \sin x (\cos y - 1) + \cos x \sin y
\]

### Step 3: Apply the identity \( \sin^2 \frac{y}{2} = \frac{1 - \cos y}{2} \)

Using the identity \( \cos y = 1 - 2 \sin^2 \frac{y}{2} \), we can rewrite \( \cos y - 1 \):

\[
\cos y - 1 = -2 \sin^2 \frac{y}{2}
\]

Substituting this into the expression:

\[
f(x+y) - f(x) = -2 \sin x \sin^2 \frac{y}{2} + \cos x \sin y
\]

### Step 4: Conclusion

We have successfully shown that:

\[
f(x+y) - f(x) = \cos x \sin y - 2 \sin x \sin^2 \left( \frac{y}{2} \right)
\]

This matches the required expression.

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Would you like further explanation on any part of this derivation?

### Relative Questions:
1. How does the identity \( \sin^2 A = \frac{1 - \cos 2A}{2} \) simplify trigonometric expressions?
2. What is the significance of expressing trigonometric identities using half-angles?
3. How can the sum of angles formula for sine be derived from first principles?
4. Why is the expression \( \cos y - 1 \) useful in this context?
5. How does the function \( \sin x \) change when a constant is added to \( x \)?
6. What are some other trigonometric identities that simplify the subtraction of functions?
7. How can this identity be used in practical applications, such as signal processing?
8. What other methods can be used to verify trigonometric identities?

### Tip:
In trigonometric identities, recognizing patterns like half-angle formulas can simplify complex expressions and make the problem more manageable.

Learn how to derive the expression f(x+y) - f(x) = cos x sin y - 2 sin x sin^2(y/2) using trigonometric identities. This problem involves the sum of angles formula for sine and the half-angle identity, suitable for advanced high school students studying trigonometry.

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