The number of solutions of equation `tanx+secx=2cosx` lying in the interval `[0, 2pi]` is

Options

A. `0`
B. `1`
C. `2`
D. `3`

Explanation

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 in terms of sine and cosine
We know that:
- \( \tan x = \frac{\sin x}{\cos x} \)
- \( \sec x = \frac{1}{\cos x} \)

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

### Step 2: Combine the left-hand side
The left-hand side can be combined over a common denominator:
\[ \frac{\sin x + 1}{\cos x} = 2 \cos x \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ \sin x + 1 = 2 \cos^2 x \]

### Step 4: Use the Pythagorean identity
Using the identity \( \cos^2 x = 1 - \sin^2 x \), we can rewrite the equation:
\[ \sin x + 1 = 2(1 - \sin^2 x) \]
This simplifies to:
\[ \sin x + 1 = 2 - 2\sin^2 x \]

### Step 5: Rearrange the equation
Rearranging gives:
\[ 2\sin^2 x + \sin x - 1 = 0 \]

### Step 6: Factor the quadratic equation
Now we can factor the quadratic:
\[ (2\sin x + 2)(\sin x - 1) = 0 \]

### Step 7: Solve for \( \sin x \)
Setting each factor to zero gives:
1. \( 2\sin x + 2 = 0 \) → \( \sin x = -1 \)
2. \( \sin x - 1 = 0 \) → \( \sin x = \frac{1}{2} \)

### Step 8: Find the solutions in the interval \( [0, 2\pi] \)
1. For \( \sin x = -1 \):
- The solution is \( x = \frac{3\pi}{2} \).

2. For \( \sin x = \frac{1}{2} \):
- The solutions are \( x = \frac{\pi}{6} \) and \( x = \frac{5\pi}{6} \).

### Step 9: Count the total solutions
The solutions in the interval \( [0, 2\pi] \) are:
- \( x = \frac{3\pi}{2} \)
- \( x = \frac{\pi}{6} \)
- \( x = \frac{5\pi}{6} \)

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

### Summary of solutions:
- \( x = \frac{3\pi}{2} \)
- \( x = \frac{\pi}{6} \)
- \( x = \frac{5\pi}{6} \)

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

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