The number of roots of the equation `tanx+secx=2cosx` in `[0, 4pi]` is

Options

A. 2
B. 4
C. 6
D. 0

Answer

To find the number of roots of the equation \( \tan x + \sec x = 2 \cos x \) in the interval \([0, 4\pi]\), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \tan x + \sec x = 2 \cos x \]
We can rewrite \(\tan x\) and \(\sec x\) in terms of sine and cosine:
\[ \frac{\sin x}{\cos x} + \frac{1}{\cos x} = 2 \cos x \]
This simplifies to:
\[ \frac{\sin x + 1}{\cos x} = 2 \cos x \]

### Step 2: Clear the fraction
Multiply both sides by \(\cos x\) (assuming \(\cos x \neq 0\)):
\[ \sin x + 1 = 2 \cos^2 x \]

### Step 3: Use the Pythagorean identity
Using the identity \(\cos^2 x = 1 - \sin^2 x\), we can substitute:
\[ \sin x + 1 = 2(1 - \sin^2 x) \]
This expands to:
\[ \sin x + 1 = 2 - 2 \sin^2 x \]

### Step 4: Rearrange the equation
Rearranging gives us:
\[ 2 \sin^2 x + \sin x - 1 = 0 \]

### Step 5: Factor the quadratic equation
We can factor this quadratic:
\[ (2 \sin x + 1)(\sin x - 1) = 0 \]
This gives us two equations to solve:
1. \(2 \sin x + 1 = 0\)
2. \(\sin x - 1 = 0\)

### Step 6: Solve for \(\sin x\)
From the first equation:
\[ 2 \sin x + 1 = 0 \implies \sin x = -\frac{1}{2} \]
From the second equation:
\[ \sin x = 1 \]

### Step 7: Find the solutions in the interval \([0, 4\pi]\)
1. For \(\sin x = -\frac{1}{2}\):
- The solutions are:
\[ x = \frac{7\pi}{6}, \frac{11\pi}{6} \quad (\text{in } [0, 2\pi]) \]
\[ x = \frac{19\pi}{6}, \frac{23\pi}{6} \quad (\text{in } [2\pi, 4\pi]) \]
- Total: 4 solutions.

2. For \(\sin x = 1\):
- The solution is:
\[ x = \frac{\pi}{2} \quad (\text{in } [0, 2\pi]) \]
\[ x = \frac{5\pi}{2} \quad (\text{in } [2\pi, 4\pi]) \]
- Total: 2 solutions.

### Step 8: Count the total number of roots
Adding the solutions together:
- From \(\sin x = -\frac{1}{2}\): 4 roots
- From \(\sin x = 1\): 2 roots

Thus, the total number of roots in the interval \([0, 4\pi]\) is:
\[ 4 + 2 = 6 \]

### Final Answer
The number of roots of the equation \( \tan x + \sec x = 2 \cos x \) in the interval \([0, 4\pi]\) is **6**.

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