If `cos A + cos^(2) A = 1` , then `sin^(2) A + sin^(4) A` is equal to

Options

A. 1
B. `1/2`
C. 0
D. `-1`

Answer

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

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

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

### Step 3: Substitute for \( \cos^2 A \)
Now, substitute \( \cos^2 A \) from Step 1 into the equation for \( \sin^2 A \):
\[ \sin^2 A = 1 - (1 - \cos A) = \cos A \]

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

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

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

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

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

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