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:

$x \in \mathbb{R}$

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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.