If `sinx+sin^(2)x=1` ,then `cos^(4)x+cos^(2)x=?`

Options

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

Answer

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

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

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

### Step 3: Substitute \( \sin^2 x \) into the equation for \( \cos^2 x \)
Substituting the expression for \( \sin^2 x \) from Step 1 into the equation for \( \cos^2 x \):
\[ \cos^2 x = 1 - (1 - \sin x) = \sin x \]

### Step 4: Substitute \( \cos^2 x \) into \( \cos^4 x + \cos^2 x \)
Now, we need to find \( \cos^4 x + \cos^2 x \):
\[ \cos^4 x + \cos^2 x = (\cos^2 x)^2 + \cos^2 x \]
Substituting \( \cos^2 x = \sin x \):
\[ = (\sin x)^2 + \sin x = \sin^2 x + \sin x \]

### Step 5: Use the original equation
From the original equation, we know that:
\[ \sin x + \sin^2 x = 1 \]
Thus, we can substitute this into our equation:
\[ \cos^4 x + \cos^2 x = 1 \]

### Conclusion
Therefore, the value of \( \cos^4 x + \cos^2 x \) is:
\[ \boxed{1} \]