To solve the equation \(\sec(x) = 2\cos(x)\), we will start by rewriting the secant function in terms of cosine:

\[
\sec(x) = \frac{1}{\cos(x)}
\]

So the equation becomes:

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

Now, multiply both sides of the equation by \(\cos(x)\) (as long as \(\cos(x) \neq 0\)) to eliminate the fraction:

\[
1 = 2\cos^2(x)
\]

Next, solve for \(\cos^2(x)\):

\[
\cos^2(x) = \frac{1}{2}
\]

Take the square root of both sides:

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

Now, solve for \(x\). The cosine function equals \(\frac{\sqrt{2}}{2}\) and \(-\frac{\sqrt{2}}{2}\) at standard angles:

1. \(\cos(x) = \frac{\sqrt{2}}{2}\) at \(x = \frac{\pi}{4} + 2n\pi\), where \(n\) is any integer.
2. \(\cos(x) = -\frac{\sqrt{2}}{2}\) at \(x = \frac{3\pi}{4} + 2n\pi\), where \(n\) is any integer.

Therefore, the general solutions are:

\[
x = \frac{\pi}{4} + 2n\pi \quad \text{or} \quad x = \frac{3\pi}{4} + 2n\pi \quad \text{for any integer } n.
\]

Would you like more details on how to derive these angles or have any questions?

### Related questions:
1. What is the domain of the cosine function?
2. How do we handle cases where \(\cos(x) = 0\) in trigonometric equations?
3. Can this solution be verified graphically?
4. How do periodic functions affect the solution of trigonometric equations?
5. What is the significance of the general solution form \(x = \theta + 2n\pi\)?

### Tip:
Remember that the secant function, \(\sec(x)\), is undefined wherever \(\cos(x) = 0\), so always consider the domain of your trigonometric functions.

Learn how to solve the trigonometric equation sec(x) = 2cos(x) by rewriting the secant function in terms of cosine and finding general solutions. This problem covers core concepts in trigonometry, including secant and cosine functions. Suitable for students in Grades 10-12.

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