Number of solution of the equation `tanx+secx=2cosx` lying in the interval `[0, 2pi]` is

Options

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

Answer

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

### Step 1: Rewrite the equation in terms of sine and cosine
We know that:
\[ \tan x = \frac{\sin x}{\cos x} \quad \text{and} \quad \sec x = \frac{1}{\cos x} \]
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
Recall that \( \cos^2 x = 1 - \sin^2 x \). Substitute this into the equation:
\[ \sin x + 1 = 2(1 - \sin^2 x) \]
Expanding this gives:
\[ \sin x + 1 = 2 - 2 \sin^2 x \]

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

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

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

### Step 8: Find the angles corresponding to the sine values
1. For \( \sin x = -\frac{1}{2} \):
- The solutions in the interval \([0, 2\pi]\) are:
\[ x = \frac{7\pi}{6}, \frac{11\pi}{6} \]
2. For \( \sin x = 1 \):
- The solution in the interval \([0, 2\pi]\) is:
\[ x = \frac{\pi}{2} \]

### Step 9: Count the total number of solutions
The total solutions in the interval \([0, 2\pi]\) are:
- From \( \sin x = -\frac{1}{2} \): 2 solutions
- From \( \sin x = 1 \): 1 solution

Thus, the total number of solutions is:
\[ 2 + 1 = 3 \]

### 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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