If `sintheta+sin^(2)theta = 1` then `cos^(2)theta+cos^(4)theta` is equal to :

Options

A. `(1)/(4)`
B. `(1)/(2)`
C. 1
D. None of these

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 given equation
We start with the equation:
\[ \sin \theta + \sin^2 \theta = 1 \]
Rearranging gives us:
\[ \sin^2 \theta = 1 - \sin \theta \]

**Hint:** You can isolate \(\sin^2 \theta\) by moving \(\sin \theta\) to the other side of the equation.

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

**Hint:** Substitute \(\sin^2 \theta\) from Step 1 into this equation.

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

**Hint:** This step connects \(\cos^2 \theta\) directly to \(\sin \theta\).

### Step 4: Squaring \(\cos^2 \theta\)
Now, we square both sides to find \(\cos^4 \theta\):
\[ \cos^4 \theta = (\cos^2 \theta)^2 = (\sin \theta)^2 = \sin^2 \theta \]

**Hint:** Remember that squaring a quantity involves multiplying it by itself.

### Step 5: Substitute \(\sin^2 \theta\) again
Now we can substitute \(\sin^2 \theta\) back into the expression we need to evaluate:
\[ \cos^2 \theta + \cos^4 \theta = \cos^2 \theta + \sin^2 \theta \]
Using the Pythagorean identity:
\[ \cos^2 \theta + \sin^2 \theta = 1 \]

**Hint:** Recognize that this is a fundamental identity in trigonometry.

### Final Answer
Thus, we find:
\[ \cos^2 \theta + \cos^4 \theta = 1 \]

So the answer is \( \boxed{1} \).