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 c sin y-2 sin x sin² (½y)

To show that \( f(x+y) - f(x) = \cos c \sin y - 2 \sin x \sin^2 \left( \frac{y}{2} \right) \) using the identity \( \sin^2 A = \frac{1 - \cos 2A}{2} \), let's start with the given function \( f(x) = \sin x \).

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

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

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

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

Factor out \( \sin x \) from the first and last terms:

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

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

The expression \( \cos y - 1 \) can be rewritten using the identity \( \cos y = 1 - 2 \sin^2 \frac{y}{2} \):

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

Substituting this into our equation:

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

### Step 4: Express the result in the desired form

Compare the result with the target form \( f(x+y) - f(x) = \cos c \sin y - 2 \sin x \sin^2 \left( \frac{y}{2} \right) \).

It matches if we assume \( \cos c = \cos x \), which means \( c = x \).

### Conclusion

Thus, the identity holds as:

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

The derivation is complete.

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Would you like further details on any step or have any questions?

### Relative Questions:
1. What are the elementary trigonometric identities used in this proof?
2. How is the sum of angles formula for sine derived?
3. Why is it useful to express \( \cos y - 1 \) in terms of \( \sin^2 \frac{y}{2} \)?
4. Can the identity \( \sin^2 A = \frac{1 - \cos 2A}{2} \) be generalized to other functions?
5. How can the identity be applied in solving trigonometric equations?
6. What is the significance of the angle \( c \) in the final expression?
7. How does the function \( f(x) = \sin x \) change with shifts in \( x \)?
8. What other trigonometric identities are useful for simplifying expressions like \( f(x+y) - f(x) \)?

### Tip:
When dealing with trigonometric identities, always consider expressing complex expressions in terms of simpler angles or half-angles to simplify the problem.

Explore the proof of the trigonometric identity f(x+y) - f(x) = cos c sin y - 2 sin x sin² (½y) using the sum of angles formula and trigonometric identities. Understand the step-by-step derivation and implications of this identity. Suitable for advanced high school students and anyone interested in trigonometry proofs.

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