If `sin A + sin^2 A = 1 ,` then the value of the expression `(cos^2A+ cos^4 A)` is

Options

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

Answer

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

### Step 1: Rearranging the equation
We start with the equation:
\[ \sin A + \sin^2 A = 1 \]
We can rearrange this to express \( \sin^2 A \):
\[ \sin^2 A = 1 - \sin A \]

### 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 \( \cos^2 A \) in terms of \( \sin^2 A \):
\[ \cos^2 A = 1 - \sin^2 A \]

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

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

### Step 5: Substitute back into the expression
Now, we substitute \( \cos^2 A \) and \( \cos^4 A \) into the expression \( \cos^2 A + \cos^4 A \):
\[ \cos^2 A + \cos^4 A = \sin A + \sin^2 A \]

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

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

So, the value of the expression \( \cos^2 A + \cos^4 A \) is \( \boxed{1} \).

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