The number of solutions of the equation `"cos"^(7) theta + "sin"^(4) theta = 1` in the interval `(-pi, pi)`, is

Options

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

Explanation

To find the number of solutions for the equation \( \cos^7 \theta + \sin^4 \theta = 1 \) in the interval \( (-\pi, \pi) \), we can follow these steps:

### Step 1: Rewrite the Equation
We start with the equation:
\[ \cos^7 \theta + \sin^4 \theta = 1 \]
Using the identity \( \sin^2 \theta = 1 - \cos^2 \theta \), we can express \( \sin^4 \theta \) as:
\[ \sin^4 \theta = (1 - \cos^2 \theta)^2 \]
Thus, we can rewrite the equation as:
\[ \cos^7 \theta + (1 - \cos^2 \theta)^2 = 1 \]

### Step 2: Expand the Equation
Now we expand \( (1 - \cos^2 \theta)^2 \):
\[ (1 - \cos^2 \theta)^2 = 1 - 2\cos^2 \theta + \cos^4 \theta \]
Substituting this back into the equation gives:
\[ \cos^7 \theta + 1 - 2\cos^2 \theta + \cos^4 \theta = 1 \]
Simplifying this, we get:
\[ \cos^7 \theta + \cos^4 \theta - 2\cos^2 \theta = 0 \]

### Step 3: Factor the Equation
We can factor out \( \cos^2 \theta \):
\[ \cos^2 \theta (\cos^5 \theta + \cos^2 \theta - 2) = 0 \]
This gives us two cases to consider:
1. \( \cos^2 \theta = 0 \)
2. \( \cos^5 \theta + \cos^2 \theta - 2 = 0 \)

### Step 4: Solve the First Case
For the first case \( \cos^2 \theta = 0 \):
\[ \cos \theta = 0 \]
The solutions for \( \theta \) in the interval \( (-\pi, \pi) \) are:
\[ \theta = -\frac{\pi}{2}, \frac{\pi}{2} \]

### Step 5: Solve the Second Case
For the second case, we set:
\[ \cos^5 \theta + \cos^2 \theta - 2 = 0 \]
Let \( x = \cos^2 \theta \). Then we have:
\[ x^2 + x - 2 = 0 \]
Factoring gives:
\[ (x - 1)(x + 2) = 0 \]
This yields:
\[ x = 1 \quad \text{or} \quad x = -2 \]
Since \( x = \cos^2 \theta \) must be non-negative, we only consider \( x = 1 \):
\[ \cos^2 \theta = 1 \implies \cos \theta = \pm 1 \]
The solutions for \( \theta \) in the interval \( (-\pi, \pi) \) are:
\[ \theta = 0, \pm \pi \]

### Step 6: Collect All Solutions
Now we combine all the solutions:
1. From \( \cos^2 \theta = 0 \): \( \theta = -\frac{\pi}{2}, \frac{\pi}{2} \)
2. From \( \cos^2 \theta = 1 \): \( \theta = 0, \pm \pi \)

### Step 7: Count Unique Solutions
The unique solutions in the interval \( (-\pi, \pi) \) are:
- \( -\frac{\pi}{2} \)
- \( 0 \)
- \( \frac{\pi}{2} \)

Thus, the total number of solutions is:
\[ \text{Total Solutions} = 3 \]

### Final Answer
The number of solutions of the equation \( \cos^7 \theta + \sin^4 \theta = 1 \) in the interval \( (-\pi, \pi) \) is \( 3 \).

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