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

Options

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

Explanation

To solve the problem, we start with the equation given:

**Step 1:** Given the equation:
\[ \sin \theta + \sin^2 \theta = 1 \]

**Step 2:** Rearranging the equation:
\[ \sin^2 \theta = 1 - \sin \theta \]

**Step 3:** We know from the Pythagorean identity that:
\[ \cos^2 \theta = 1 - \sin^2 \theta \]

**Step 4:** Substituting \( \sin^2 \theta \) from Step 2 into the Pythagorean identity:
\[ \cos^2 \theta = 1 - (1 - \sin \theta) = \sin \theta \]

**Step 5:** Now we need to find \( \cos^4 \theta \):
\[ \cos^4 \theta = (\cos^2 \theta)^2 = (\sin \theta)^2 = \sin^2 \theta \]

**Step 6:** We now want to find \( \cos^2 \theta + \cos^4 \theta \):
\[ \cos^2 \theta + \cos^4 \theta = \sin \theta + \sin^2 \theta \]

**Step 7:** From the original equation, we know that:
\[ \sin \theta + \sin^2 \theta = 1 \]

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

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