To rewrite the expression using the double-angle formula, let's start by analyzing the given expression:

$\frac{1}{2} \cos^2 x - \frac{1}{4}$

We want to apply a double-angle identity. The relevant double-angle formula for cosine is:

$\cos(2x) = 2\cos^2(x) - 1$

We can rearrange this identity to express $\cos^2(x)$ in terms of $\cos(2x)$:

$\cos^2(x) = \frac{1 + \cos(2x)}{2}$

Now, substitute this into the original expression:

$\frac{1}{2} \cos^2 x - \frac{1}{4} = \frac{1}{2} \left(\frac{1 + \cos(2x)}{2}\right) - \frac{1}{4}$

Simplify:

$= \frac{1}{4} (1 + \cos(2x)) - \frac{1}{4}$

Now, combine the constants:

$= \frac{1}{4} \cos(2x) + \frac{1}{4} - \frac{1}{4}$

$= \frac{1}{4} \cos(2x)$

So, the expression rewritten using the double-angle formula is:

$\frac{1}{4} \cos(2x)$

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Would you like any more details on this process, or do you have any other questions?

Here are 5 related questions to explore:

1. What are the other trigonometric double-angle identities?
2. How can the identity for $\sin(2x)$ be derived?
3. How can you apply the double-angle formula to simplify $\sin^2 x$?
4. What is the general method for simplifying trigonometric expressions using identities?
5. How do half-angle formulas relate to double-angle formulas?

Tip: Always try to identify which trigonometric identities apply to your expression early on to simplify the process.