The number of solutions of the equation `"tan" x + "sec"x = 2"cos" x` lying in the interval `[0, 2 pi]` is

Options

A. 0
B. 2
C. 4
D. 8

Answer

To find the number of solutions of the equation \( \tan x + \sec x = 2 \cos x \) in the interval \([0, 2\pi]\), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \tan x + \sec x = 2 \cos x \]
We can express \(\tan x\) and \(\sec x\) in terms of sine and cosine:
\[ \frac{\sin x}{\cos x} + \frac{1}{\cos x} = 2 \cos x \]
Combining the left side gives:
\[ \frac{\sin x + 1}{\cos x} = 2 \cos x \]

### Step 2: Cross-multiply
Cross-multiplying to eliminate the fraction:
\[ \sin x + 1 = 2 \cos^2 x \]

### Step 3: Use the Pythagorean identity
Using the identity \( \cos^2 x = 1 - \sin^2 x \):
\[ \sin x + 1 = 2(1 - \sin^2 x) \]
Expanding the right side:
\[ \sin x + 1 = 2 - 2\sin^2 x \]

### Step 4: Rearranging the equation
Rearranging gives us a quadratic equation in terms of \(\sin x\):
\[ 2\sin^2 x + \sin x - 1 = 0 \]

### Step 5: Solve the quadratic equation
We can solve this quadratic equation using the quadratic formula:
\[ \sin x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 2\), \(b = 1\), and \(c = -1\):
\[ \sin x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 2 \cdot (-1)}}{2 \cdot 2} \]
Calculating the discriminant:
\[ \sin x = \frac{-1 \pm \sqrt{1 + 8}}{4} = \frac{-1 \pm 3}{4} \]
This gives us two potential solutions:
1. \( \sin x = \frac{2}{4} = \frac{1}{2} \)
2. \( \sin x = \frac{-4}{4} = -1 \)

### Step 6: Find the values of \(x\)
1. For \( \sin x = \frac{1}{2} \):
- The solutions in the interval \([0, 2\pi]\) are:
\[ x = \frac{\pi}{6}, \quad x = \frac{5\pi}{6} \]
2. For \( \sin x = -1 \):
- The solution in the interval \([0, 2\pi]\) is:
\[ x = \frac{3\pi}{2} \]

### Step 7: Count the total solutions
Combining the solutions:
- From \( \sin x = \frac{1}{2} \): 2 solutions
- From \( \sin x = -1 \): 1 solution

Thus, the total number of solutions in the interval \([0, 2\pi]\) is:
\[ \text{Total solutions} = 2 + 1 = 3 \]

### Final Answer
The number of solutions of the equation \( \tan x + \sec x = 2 \cos x \) lying in the interval \([0, 2\pi]\) is **3**.

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