If ` sin theta + sin ^(2) theta =1, " then" cos^(2) theta + cos^(4) theta `

Options

A. 1
B. `sqrt(2)`
C. 0
D. 2

Answer

To solve the equation \( \sin \theta + \sin^2 \theta = 1 \) and find the value of \( \cos^2 \theta + \cos^4 \theta \), we can follow these steps:

### Step 1: Rearranging the Equation
Start with the given equation:
\[ \sin \theta + \sin^2 \theta = 1 \]
Rearranging gives:
\[ \sin^2 \theta = 1 - \sin \theta \]

### Step 2: Using the Pythagorean Identity
We know from the Pythagorean identity that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Thus, we can express \( \cos^2 \theta \) in terms of \( \sin^2 \theta \):
\[ \cos^2 \theta = 1 - \sin^2 \theta \]

### Step 3: Substitute \( \sin^2 \theta \)
From Step 1, we have \( \sin^2 \theta = 1 - \sin \theta \). Substitute this into the equation for \( \cos^2 \theta \):
\[ \cos^2 \theta = 1 - (1 - \sin \theta) = \sin \theta \]

### Step 4: Finding \( \cos^4 \theta \)
Now, we need to find \( \cos^4 \theta \):
\[ \cos^4 \theta = (\cos^2 \theta)^2 = (\sin \theta)^2 = \sin^2 \theta \]

### Step 5: Combine \( \cos^2 \theta \) and \( \cos^4 \theta \)
Now we can find \( \cos^2 \theta + \cos^4 \theta \):
\[ \cos^2 \theta + \cos^4 \theta = \sin \theta + \sin^2 \theta \]

### Step 6: Substitute Back to Original Equation
From the original equation, we know:
\[ \sin \theta + \sin^2 \theta = 1 \]
Thus:
\[ \cos^2 \theta + \cos^4 \theta = 1 \]

### Final Answer
The value of \( \cos^2 \theta + \cos^4 \theta \) is:
\[ \boxed{1} \]