The number of solutions of x in the interval `[-pi, pi]` of the equation `(1+cot267^(@)) (1+tan222^(@)) = sec^(2)x + cos^(2)x` is

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the equation \( (1 + \cot 267^\circ)(1 + \tan 222^\circ) = \sec^2 x + \cos^2 x \) for the number of solutions in the interval \([- \pi, \pi]\), we will follow these steps:

### Step 1: Simplify the Left-Hand Side
First, we need to simplify the left-hand side of the equation:

\[ 1 + \cot 267^\circ = 1 + \cot(270^\circ - 3^\circ) = 1 + \tan 3^\circ \]
\[ 1 + \tan 222^\circ = 1 + \tan(270^\circ - 48^\circ) = 1 + \cot 48^\circ \]

### Step 2: Substitute Cotangent
Using the identity \( \cot \theta = \frac{1}{\tan \theta} \), we can express \( \cot 48^\circ \):

\[ \cot 48^\circ = \frac{1}{\tan 48^\circ} \]

### Step 3: Rewrite the Left-Hand Side
Now we can rewrite the left-hand side:

\[ (1 + \tan 3^\circ)(1 + \cot 48^\circ) = (1 + \tan 3^\circ)\left(1 + \frac{1}{\tan 48^\circ}\right) \]

### Step 4: Simplify Further
Using the identity for \( \tan(a + b) \):

\[ 1 + \cot 48^\circ = 1 + \frac{1}{\tan 48^\circ} = \frac{\tan 48^\circ + 1}{\tan 48^\circ} \]

Thus, the left-hand side becomes:

\[ (1 + \tan 3^\circ) \cdot \frac{\tan 48^\circ + 1}{\tan 48^\circ} \]

### Step 5: Calculate the Left-Hand Side
Now we can calculate the left-hand side further, but we already know that it simplifies to a constant value.

### Step 6: Set Up the Right-Hand Side
Now we need to analyze the right-hand side:

\[ \sec^2 x + \cos^2 x = \frac{1}{\cos^2 x} + \cos^2 x \]

Let \( t = \cos^2 x \). Then we can rewrite the equation as:

\[ \frac{1}{t} + t = C \quad \text{(where \( C \) is the value of the left-hand side)} \]

### Step 7: Multiply by \( t \)
Multiplying through by \( t \) gives:

\[ 1 + t^2 = Ct \]

Rearranging gives:

\[ t^2 - Ct + 1 = 0 \]

### Step 8: Find the Discriminant
To find the number of solutions for \( t \), we calculate the discriminant:

\[ D = C^2 - 4 \]

### Step 9: Determine the Number of Solutions
The number of solutions for \( t \) depends on the discriminant:

- If \( D > 0 \): Two distinct solutions.
- If \( D = 0 \): One solution.
- If \( D < 0 \): No real solutions.

### Step 10: Find \( t \) Values
Since \( t = \cos^2 x \), we need to check the solutions in the interval \([- \pi, \pi]\).

### Step 11: Count the Solutions
If \( t = 1 \), then \( \cos^2 x = 1 \) implies \( x = 0, \pi, -\pi \). This gives us three solutions in the interval \([- \pi, \pi]\).

### Final Answer
Thus, the number of solutions of \( x \) in the interval \([- \pi, \pi]\) is **3**.

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