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 solve 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 know that \( \sin^2 \theta = 1 - \cos^2 \theta \). Therefore, we can express \( \sin^4 \theta \) as:
\[ \sin^4 \theta = (1 - \cos^2 \theta)^2 = 1 - 2\cos^2 \theta + \cos^4 \theta \]
Substituting this into the original equation gives:
\[ \cos^7 \theta + (1 - 2\cos^2 \theta + \cos^4 \theta) = 1 \]

### Step 2: Simplify the equation
Now, we simplify the equation:
\[ \cos^7 \theta + 1 - 2\cos^2 \theta + \cos^4 \theta = 1 \]
Subtracting 1 from both sides:
\[ \cos^7 \theta + \cos^4 \theta - 2\cos^2 \theta = 0 \]

### 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 \( \cos^2 \theta = 0 \):
\[ \cos \theta = 0 \implies \theta = \pm \frac{\pi}{2} \]

### Step 5: Solve the second case
Now, we need to solve \( \cos^5 \theta + \cos^2 \theta - 2 = 0 \). Let \( x = \cos^2 \theta \):
\[ x^2 + x - 2 = 0 \]
Factoring gives:
\[ (x - 1)(x + 2) = 0 \]
Thus, \( x = 1 \) or \( x = -2 \). Since \( x = \cos^2 \theta \) must be non-negative, we only consider \( x = 1 \):
\[ \cos^2 \theta = 1 \implies \cos \theta = \pm 1 \implies \theta = 0, \pm \pi \]

### Step 6: Collect all solutions
Now we have the following solutions in the interval \( (-\pi, \pi) \):
1. \( \theta = -\frac{\pi}{2} \)
2. \( \theta = 0 \)
3. \( \theta = \frac{\pi}{2} \)

### Step 7: Count the solutions
The solutions we found are:
- \( \theta = -\frac{\pi}{2} \)
- \( \theta = 0 \)
- \( \theta = \frac{\pi}{2} \)

Thus, there are a total of **3 solutions** in the interval \( (-\pi, \pi) \).

### 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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