\cos (x-(\pi )/(4))-\cos (x+(\pi )/(4))=\sqrt(2)\sin x

We are tasked with solving the following trigonometric equation:

\[
\cos \left( x - \frac{\pi}{4} \right) - \cos \left( x + \frac{\pi}{4} \right) = \sqrt{2} \sin x
\]

### Step 1: Use the cosine difference identity

We can apply the following identity for the difference of cosines:
\[
\cos A - \cos B = -2 \sin \left( \frac{A+B}{2} \right) \sin \left( \frac{A-B}{2} \right)
\]

Let \( A = x - \frac{\pi}{4} \) and \( B = x + \frac{\pi}{4} \). Using this identity, we get:

\[
\cos \left( x - \frac{\pi}{4} \right) - \cos \left( x + \frac{\pi}{4} \right) = -2 \sin \left( \frac{\left(x - \frac{\pi}{4}\right) + \left(x + \frac{\pi}{4}\right)}{2} \right) \sin \left( \frac{\left(x - \frac{\pi}{4}\right) - \left(x + \frac{\pi}{4}\right)}{2} \right)
\]

### Step 2: Simplify the terms

The first term becomes:
\[
\frac{\left(x - \frac{\pi}{4}\right) + \left(x + \frac{\pi}{4}\right)}{2} = \frac{2x}{2} = x
\]
The second term becomes:
\[
\frac{\left(x - \frac{\pi}{4}\right) - \left(x + \frac{\pi}{4}\right)}{2} = \frac{-\frac{\pi}{2}}{2} = -\frac{\pi}{4}
\]

Thus, the equation simplifies to:
\[
-2 \sin x \sin \left( -\frac{\pi}{4} \right) = \sqrt{2} \sin x
\]

### Step 3: Simplify further

Since \( \sin \left( -\frac{\pi}{4} \right) = -\frac{\sqrt{2}}{2} \), we can substitute this into the equation:

\[
-2 \sin x \cdot \left( -\frac{\sqrt{2}}{2} \right) = \sqrt{2} \sin x
\]

This simplifies to:

\[
\sqrt{2} \sin x = \sqrt{2} \sin x
\]

### Step 4: Conclusion

The equation is an identity, meaning it is true for all values of \( x \). Therefore, the solution is:

\[
x \in \mathbb{R}
\]

---

Would you like more details on any step or have any specific questions?

### Related Questions:
1. How can we apply the sum-to-product identities to other trigonometric equations?
2. What are some real-world applications of trigonometric identities?
3. How do you solve a trigonometric equation involving both sine and cosine functions?
4. How can you derive the cosine and sine difference identities?
5. What is the general solution to \( \sin x = 0 \) and \( \cos x = 0 \)?

**Tip:** Always double-check your trigonometric identities and ensure you apply the correct one when simplifying equations.

\cos(x - \frac{\pi}{4}) - \cos(x + \frac{\pi}{4}) = \sqrt{2} \sin x

This detailed solution shows how to solve the trigonometric equation \cos(x - \frac{\pi}{4}) - \cos(x + \frac{\pi}{4}) = \sqrt{2} \sin x using the cosine difference identity. Key concepts include trigonometric identities and simplification techniques, ideal for students in Grades 10-12.

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