Find the number of distinct solution of ` sec x + tan x =sqrt(3)` , where ` 0 le x le 3pi`

Answer

To solve the equation \( \sec x + \tan x = \sqrt{3} \) for \( 0 \leq x \leq 3\pi \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \sec x + \tan x = \sqrt{3} \]
We know that:
\[ \sec x = \frac{1}{\cos x} \quad \text{and} \quad \tan x = \frac{\sin x}{\cos x} \]
Thus, we can rewrite the equation as:
\[ \frac{1 + \sin x}{\cos x} = \sqrt{3} \]

### Step 2: Cross-multiply
Cross-multiplying gives us:
\[ 1 + \sin x = \sqrt{3} \cos x \]

### Step 3: Rearranging the equation
Rearranging this equation, we get:
\[ \sqrt{3} \cos x - \sin x = 1 \]

### Step 4: Use the identity
We can use the identity \( \sqrt{3} \cos x - \sin x = 1 \) to express it in terms of a single trigonometric function. We can rewrite this as:
\[ \sqrt{3} \cos x - \sin x = 1 \]
This can be transformed into the form:
\[ R \cos(x + \phi) = 1 \]
where \( R = \sqrt{(\sqrt{3})^2 + (-1)^2} = \sqrt{3 + 1} = 2 \) and \( \phi = \tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) = -\frac{\pi}{6} \).

### Step 5: Solve for \( x \)
So we have:
\[ 2 \cos\left(x - \frac{\pi}{6}\right) = 1 \]
Dividing both sides by 2 gives:
\[ \cos\left(x - \frac{\pi}{6}\right) = \frac{1}{2} \]

### Step 6: Find the general solutions
The general solutions for \( \cos \theta = \frac{1}{2} \) are:
\[ \theta = \frac{\pi}{3} + 2k\pi \quad \text{and} \quad \theta = -\frac{\pi}{3} + 2k\pi \]
Substituting back for \( x \):
1. \( x - \frac{\pi}{6} = \frac{\pi}{3} + 2k\pi \)
\[ x = \frac{\pi}{3} + \frac{\pi}{6} + 2k\pi = \frac{2\pi}{6} + \frac{\pi}{6} + 2k\pi = \frac{3\pi}{6} + 2k\pi = \frac{\pi}{2} + 2k\pi \]

2. \( x - \frac{\pi}{6} = -\frac{\pi}{3} + 2k\pi \)
\[ x = -\frac{\pi}{3} + \frac{\pi}{6} + 2k\pi = -\frac{2\pi}{6} + \frac{\pi}{6} + 2k\pi = -\frac{\pi}{6} + 2k\pi \]

### Step 7: Find distinct solutions in the interval \( [0, 3\pi] \)
Now we will find the distinct solutions for \( k = 0, 1, 2 \):
1. For \( x = \frac{\pi}{2} + 2k\pi \):
- \( k = 0 \): \( x = \frac{\pi}{2} \)
- \( k = 1 \): \( x = \frac{\pi}{2} + 2\pi = \frac{5\pi}{2} \)
- \( k = 2 \): \( x = \frac{\pi}{2} + 4\pi = \frac{9\pi}{2} \) (not in the range)

2. For \( x = -\frac{\pi}{6} + 2k\pi \):
- \( k = 0 \): \( x = -\frac{\pi}{6} \) (not in the range)
- \( k = 1 \): \( x = -\frac{\pi}{6} + 2\pi = \frac{11\pi}{6} \)
- \( k = 2 \): \( x = -\frac{\pi}{6} + 4\pi = \frac{23\pi}{6} \) (not in the range)

### Conclusion
The distinct solutions in the interval \( [0, 3\pi] \) are:
1. \( x = \frac{\pi}{2} \)
2. \( x = \frac{5\pi}{2} \)
3. \( x = \frac{11\pi}{6} \)

Thus, the number of distinct solutions is **3**.