The number of solutions of the equation ` 1 +sin^(4) x = cos ^(2) 3x, x in [-(5pi)/(2),(5pi)/(2)]` is

Options

A. 7
B. 3
C. 4
D. 5

Explanation

To solve the equation \( 1 + \sin^4 x = \cos^2 3x \) for \( x \) in the interval \( \left[-\frac{5\pi}{2}, \frac{5\pi}{2}\right] \), we will follow these steps:

### Step 1: Analyze the ranges of both sides of the equation

The left-hand side of the equation is \( 1 + \sin^4 x \). Since \( \sin^4 x \) is always non-negative and reaches a maximum value of 1 (when \( \sin x = \pm 1 \)), we can conclude that:
\[ 1 + \sin^4 x \geq 1 \]
Thus, the minimum value of the left-hand side is 1.

The right-hand side is \( \cos^2 3x \). Since \( \cos^2 3x \) also ranges from 0 to 1, we have:
\[ 0 \leq \cos^2 3x \leq 1 \]

### Step 2: Set up the equation

From the analysis, we can see that for the equation \( 1 + \sin^4 x = \cos^2 3x \) to hold, we must have:
\[ 1 + \sin^4 x \geq 1 \quad \text{and} \quad \cos^2 3x \leq 1 \]
This implies that the only possible equality occurs when:
\[ 1 + \sin^4 x = 1 \implies \sin^4 x = 0 \]
Thus, we have:
\[ \sin x = 0 \]

### Step 3: Find the solutions for \( \sin x = 0 \)

The solutions to \( \sin x = 0 \) are given by:
\[ x = n\pi \quad \text{where } n \text{ is an integer.} \]

### Step 4: Determine the integer values of \( n \) within the specified interval

We need to find the integer values of \( n \) such that:
\[ -\frac{5\pi}{2} \leq n\pi \leq \frac{5\pi}{2} \]
Dividing the entire inequality by \( \pi \):
\[ -\frac{5}{2} \leq n \leq \frac{5}{2} \]
The integer values of \( n \) that satisfy this inequality are:
\[ n = -2, -1, 0, 1, 2 \]

### Step 5: List the corresponding \( x \) values

Now, substituting these \( n \) values back into \( x = n\pi \):
- For \( n = -2 \): \( x = -2\pi \)
- For \( n = -1 \): \( x = -\pi \)
- For \( n = 0 \): \( x = 0 \)
- For \( n = 1 \): \( x = \pi \)
- For \( n = 2 \): \( x = 2\pi \)

### Step 6: Count the solutions

The solutions we have found are:
- \( -2\pi \)
- \( -\pi \)
- \( 0 \)
- \( \pi \)
- \( 2\pi \)

Thus, there are a total of 5 solutions.

### Final Answer

The number of solutions of the equation \( 1 + \sin^4 x = \cos^2 3x \) in the interval \( \left[-\frac{5\pi}{2}, \frac{5\pi}{2}\right] \) is \( \boxed{5} \).

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