The given equation is:

\[
\cos(x) = \frac{1}{2} \sqrt{2 + \sqrt{3}}
\]

To solve this, let's determine the value of \( x \) that satisfies the equation.

### Step 1: Simplify the Right Side
Let's check if we can simplify the expression \( \frac{1}{2} \sqrt{2 + \sqrt{3}} \).

First, note that \( \cos(x) \) corresponds to certain angles, and its values can often be related to known angles. However, the expression is not immediately recognizable as a standard cosine value. Let's square both sides to see if it simplifies.

\[
\cos^2(x) = \left(\frac{1}{2} \sqrt{2 + \sqrt{3}}\right)^2
\]

\[
\cos^2(x) = \frac{1}{4} (2 + \sqrt{3})
\]

### Step 2: Find Possible Cosine Values
The equation now becomes:

\[
\cos^2(x) = \frac{1}{4} (2 + \sqrt{3})
\]

This implies:

\[
\cos(x) = \pm \frac{\sqrt{2 + \sqrt{3}}}{2}
\]

We now need to determine the angles \( x \) that satisfy this condition. These angles might not correspond to standard angles, so we typically express the solution as:

\[
x = \cos^{-1}\left(\frac{\sqrt{2 + \sqrt{3}}}{2}\right) \quad \text{or} \quad x = \cos^{-1}\left(-\frac{\sqrt{2 + \sqrt{3}}}{2}\right)
\]

### Final Answer
The general solutions for \( x \) are:

\[
x = 2n\pi \pm \cos^{-1}\left(\frac{\sqrt{2 + \sqrt{3}}}{2}\right), \quad n \in \mathbb{Z}
\]

Let me know if you need further details or if you have any questions!

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Here are 5 related questions:
1. What are the properties of the cosine function?
2. How do we derive the double-angle and half-angle identities for cosine?
3. Can we express the expression \( \sqrt{2 + \sqrt{3}} \) in a simpler trigonometric form?
4. What are the general solutions to \( \cos(x) = a \) for any value of \( a \)?
5. How does the unit circle help in understanding the solutions to cosine equations?

**Tip:** When dealing with trigonometric equations, always consider the unit circle and the symmetry of the trigonometric functions to find all possible solutions.

Learn how to solve the trigonometric equation \( \cos(x) = \frac{1}{2} \sqrt{2 + \sqrt{3}} \) step-by-step with detailed explanations. Explore the inverse cosine formula and double-angle identities for cosine. Suitable for advanced high school students and those interested in trigonometric equations.

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