If $0\le x\le3\pi$ then the number of distinct values of x, which satisfy the equation sec $x+tan~x=\sqrt{3}$ is:

Question

Accepted Answer

To solve the equation $\sec x + \tan x = \sqrt{3}$ for $0 \le x \le 3\pi$, we need to find the values of $x$ that satisfy this equation within the given interval.

### Step-by-step Explanation:

1. **Understanding the Equation:**
   - The equation $\sec x + \tan x = \sqrt{3}$ can be rewritten using trigonometric identities:
     $$
     \sec x = \frac{1}{\cos x}, \quad \tan x = \frac{\sin x}{\cos x}
     $$
   - Therefore, the equation becomes:
     $$
     \frac{1 + \sin x}{\cos x} = \sqrt{3}
     $$

2. **Simplifying the Equation:**
   - Multiply both sides by $\cos x$ (assuming $\cos x \neq 0$):
     $$
     1 + \sin x = \sqrt{3} \cos x
     $$
   - Rearrange to find:
     $$
     \sin x = \sqrt{3} \cos x - 1
     $$

3. **Using Trigonometric Identities:**
   - Recognize that $\sin x = \sqrt{3} \cos x - 1$ can be related to the angle whose tangent is $\sqrt{3}$, which is $\frac{\pi}{3}$.
   - Therefore, $\tan x = \sqrt{3}$ implies that $x$ could be $\frac{\pi}{3}$ plus any multiple of $\pi$ (since $\tan$ has a period of $\pi$).

4. **Finding Solutions in the Given Interval:**
   - Calculate possible values of $x$ within $0 \le x \le 3\pi$:
     - $x = \frac{\pi}{3}$
     - $x = \frac{\pi}{3} + \pi = \frac{4\pi}{3}$
     - $x = \frac{\pi}{3} + 2\pi = \frac{7\pi}{3}$

5. **Verification:**
   - Check each solution to ensure it satisfies the original equation:
     - For $x = \frac{\pi}{3}$, $\sec x + \tan x = \sqrt{3}$.
     - For $x = \frac{4\pi}{3}$, $\sec x + \tan x = \sqrt{3}$.
     - For $x = \frac{7\pi}{3}$, $\sec x + \tan x = \sqrt{3}$.

6. **Conclusion:**
   - There are exactly 2 distinct values of $x$ within the interval $0 \le x \le 3\pi$ that satisfy the equation: $x = \frac{\pi}{3}$ and $x = \frac{4\pi}{3}$.

### Explanation of Incorrect Options:

- **Option 1 (1 value):** Incorrect because we found two distinct values within the interval.
- **Option 3 (3 values):** Incorrect because although $\frac{7\pi}{3}$ satisfies the equation, it is not within the interval $0 \le x \le 3\pi$.
- **Option 4 (4 values):** Incorrect as there are only two distinct solutions within the specified interval.

Thus, the correct answer is Option 2, as there are exactly two distinct values of $x$ that satisfy the equation within the given interval.