Choose the correct answer in each of the following questions:
If `sinA + sin^(2)A=1 " then " cos^(2)A+cos^(4)A=?`

Options

A. `(1)/(2)`
B. 1
C. 2
D. 3

Explanation

To solve the problem, we start with the given equation and work step by step to find the value of \( \cos^2 A + \cos^4 A \).

### Step 1: Start with the given equation
We have:
\[ \sin A + \sin^2 A = 1 \]

### Step 2: Rearrange the equation
We can rearrange this equation to express \( \sin A \) in terms of \( \sin^2 A \):
\[ \sin^2 A = 1 - \sin A \]

### Step 3: Substitute \( \sin^2 A \) with \( \cos^2 A \)
Using the Pythagorean identity \( \sin^2 A + \cos^2 A = 1 \), we can express \( \sin^2 A \) as:
\[ \sin^2 A = 1 - \cos^2 A \]
Substituting this into our rearranged equation gives:
\[ \sin A = 1 - (1 - \cos^2 A) = \cos^2 A \]

### Step 4: Substitute \( \sin A \) into the expression we want to find
Now we need to find \( \cos^2 A + \cos^4 A \). We can express \( \cos^4 A \) as \( (\cos^2 A)^2 \):
\[ \cos^2 A + \cos^4 A = \cos^2 A + (\cos^2 A)^2 \]
Let \( x = \cos^2 A \), then we have:
\[ x + x^2 \]

### Step 5: Substitute \( x \) with \( \sin A \)
From our earlier steps, we know that \( \sin A = \cos^2 A \). Thus, we can replace \( x \) with \( \sin A \):
\[ \sin A + \sin^2 A \]

### Step 6: Use the original equation
From the original equation, we know that:
\[ \sin A + \sin^2 A = 1 \]

### Conclusion
Thus, we find that:
\[ \cos^2 A + \cos^4 A = 1 \]

### Final Answer
The value of \( \cos^2 A + \cos^4 A \) is \( 1 \).

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