2sec x = tan x + cot x	solve for x when 0<=x<=2\pi

To solve the equation \( 2\sec(x) = \tan(x) + \cot(x) \) for \( x \) in the interval \( 0 \leq x \leq 2\pi \), let's proceed step by step.

### Step 1: Simplify the equation
First, recall the following trigonometric identities:
\[
\sec(x) = \frac{1}{\cos(x)}, \quad \tan(x) = \frac{\sin(x)}{\cos(x)}, \quad \cot(x) = \frac{\cos(x)}{\sin(x)}
\]

Substituting these into the equation:
\[
2\cdot\frac{1}{\cos(x)} = \frac{\sin(x)}{\cos(x)} + \frac{\cos(x)}{\sin(x)}
\]

### Step 2: Multiply through by \(\cos(x)\)
To eliminate the denominator \(\cos(x)\), multiply every term by \(\cos(x)\):
\[
2 = \sin(x) + \frac{\cos^2(x)}{\sin(x)}
\]

### Step 3: Multiply through by \(\sin(x)\)
To simplify further, multiply the entire equation by \(\sin(x)\):
\[
2\sin(x) = \sin^2(x) + \cos^2(x)
\]

Using the Pythagorean identity \(\sin^2(x) + \cos^2(x) = 1\), the equation becomes:
\[
2\sin(x) = 1
\]

### Step 4: Solve for \(\sin(x)\)
\[
\sin(x) = \frac{1}{2}
\]

### Step 5: Find the values of \( x \)
The general solution for \(\sin(x) = \frac{1}{2}\) in the interval \(0 \leq x \leq 2\pi\) occurs at:
\[
x = \frac{\pi}{6}, \quad x = \frac{5\pi}{6}
\]

### Final Answer:
The solutions to the equation \( 2\sec(x) = \tan(x) + \cot(x) \) within the interval \( 0 \leq x \leq 2\pi \) are:
\[
x = \frac{\pi}{6}, \quad x = \frac{5\pi}{6}
\]

Would you like more details on any of these steps, or have any other questions?

Here are some related questions you might explore:
1. What is the general solution for \( \sin(x) = \frac{1}{2} \)?
2. How can you derive the identity \( \sin^2(x) + \cos^2(x) = 1 \)?
3. What is the domain and range of the secant function?
4. How do you solve trigonometric equations involving multiple angles?
5. What is the significance of the interval \( 0 \leq x \leq 2\pi \) in trigonometric equations?

**Tip:** Always remember to check if the solutions fall within the given interval when solving trigonometric equations.

Learn how to solve the trigonometric equation 2sec(x) = tan(x) + cot(x) step-by-step. This problem involves trigonometric identities and equations, specifically addressing the secant, tangent, and cotangent functions. Suitable for advanced high school students and those studying trigonometry.

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