To solve the trigonometric equation \(\cos(30^\circ + x) - \sin(x) = 0\), we will use trigonometric identities and solve for \(x\). Here’s the step-by-step process:

1. **Rewrite the equation**:

   \[
   \cos(30^\circ + x) - \sin(x) = 0
   \]

   This can be rearranged to:

   \[
   \cos(30^\circ + x) = \sin(x)
   \]

2. **Use a trigonometric identity**:

   Recall that \(\sin(x) = \cos(90^\circ - x)\). This identity will help us express \(\sin(x)\) in terms of cosine:

   \[
   \cos(30^\circ + x) = \cos(90^\circ - x)
   \]

3. **Set up the equation using the identity for cosine**:

   For two cosine expressions to be equal, the angles must either be the same or differ by \(360^\circ\), or they must be supplementary (i.e., their sum is \(180^\circ\)):

   \[
   30^\circ + x = 90^\circ - x + 360^\circ n
   \]
   or
   \[
   30^\circ + x = 270^\circ + x + 360^\circ n
   \]
   where \(n\) is an integer.

4. **Solve the first equation for \(x\)**:

   \[
   30^\circ + x = 90^\circ - x + 360^\circ n
   \]

   Combine like terms:

   \[
   2x = 60^\circ + 360^\circ n
   \]

   \[
   x = 30^\circ + 180^\circ n
   \]

5. **Solve the second equation for \(x\)**:

   \[
   30^\circ + x = 270^\circ + x + 360^\circ n
   \]

   Combine like terms:

   \[
   0 = 240^\circ + 360^\circ n
   \]

   Since \(x\) cancels out, this equation does not provide a valid solution for \(x\).

6. **General solution for \(x\)**:

   The valid general solution is:

   \[
   x = 30^\circ + 180^\circ n, \quad \text{where } n \text{ is any integer.}
   \]

So, the solution to the equation \(\cos(30^\circ + x) - \sin(x) = 0\) is \(x = 30^\circ + 180^\circ n\).

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

### Related Questions:
1. How do you solve trigonometric equations involving both sine and cosine?
2. What are the general solutions for equations involving sine and cosine?
3. How can you graphically interpret solutions to trigonometric equations?
4. What are some strategies for solving more complex trigonometric equations?
5. How do the unit circle and reference angles help solve trigonometric equations?

### Tip:
Trigonometric identities are powerful tools for simplifying equations. Knowing key identities like \(\sin(x) = \cos(90^\circ - x)\) and others can greatly help in solving equations.

Learn how to solve the trigonometric equation cos(30º + x) - sin(x) = 0 step-by-step. This problem involves using trigonometric identities and solving equations. Suitable for advanced high school students or those studying trigonometry.

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