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. 1
C. 2
D. 3

Answer

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

### Step 1: Rewrite the equation
Start by rewriting the equation using the definitions of tangent and secant:
\[ \tan x + \sec x = \frac{\sin x}{\cos x} + \frac{1}{\cos x} = \frac{\sin x + 1}{\cos x} \]
Thus, the equation becomes:
\[ \frac{\sin x + 1}{\cos x} = 2 \cos x \]

### 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
Recall the identity \( \cos^2 x = 1 - \sin^2 x \). Substitute this into the equation:
\[ \sin x + 1 = 2(1 - \sin^2 x) \]
This simplifies to:
\[ \sin x + 1 = 2 - 2\sin^2 x \]

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

### Step 5: Solve the quadratic equation
Now, we can solve the quadratic equation \( 2\sin^2 x + \sin x - 1 = 0 \) using the quadratic formula:
\[ \sin x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 2 \cdot (-1)}}{2 \cdot 2} \]
Calculating the discriminant:
\[ 1 + 8 = 9 \]
So, we have:
\[ \sin x = \frac{-1 \pm 3}{4} \]
This gives us two 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:
\[ x = \frac{\pi}{6}, \frac{5\pi}{6} \]

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

### Step 7: Count the solutions
Thus, the total number of solutions in the interval \( [0, 2\pi] \) is:
- From \( \sin x = \frac{1}{2} \): 2 solutions
- From \( \sin x = -1 \): 1 solution

Therefore, the total number of solutions 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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