If `sintheta=cos^(2)theta`, then what is `cos^(2)theta(1+cos^(2)theta)` equal to ?

Options

A. 1
B. 0
C. `cos^(2)theta`
D. `2sintheta`

Explanation

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

### Step 1: Rewrite the expression
We start with the expression we need to evaluate:
\[ \cos^2 \theta (1 + \cos^2 \theta) \]
This can be expanded as:
\[ \cos^2 \theta + \cos^4 \theta \]

**Hint:** Remember that when you see a product like this, you can distribute the terms to simplify the expression.

### Step 2: Use the given equation
From the problem, we know that:
\[ \sin \theta = \cos^2 \theta \]
We can square both sides to relate \( \sin^2 \theta \) and \( \cos^4 \theta \):
\[ \sin^2 \theta = (\cos^2 \theta)^2 = \cos^4 \theta \]

**Hint:** Squaring both sides of an equation can help you find relationships between different trigonometric functions.

### Step 3: Substitute \( \sin^2 \theta \)
Now, we can substitute \( \cos^4 \theta \) in our expression:
\[ \cos^2 \theta + \cos^4 \theta = \cos^2 \theta + \sin^2 \theta \]

**Hint:** Look for ways to substitute known values or identities to simplify your expression.

### Step 4: Use the Pythagorean identity
We know from the Pythagorean identity that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Thus, we can rewrite our expression:
\[ \cos^2 \theta + \sin^2 \theta = 1 \]

**Hint:** The Pythagorean identity is a key tool in trigonometry that can simplify many expressions.

### Step 5: Conclusion
Therefore, we find that:
\[ \cos^2 \theta (1 + \cos^2 \theta) = 1 \]

**Final Answer:**
\[ \cos^2 \theta (1 + \cos^2 \theta) = 1 \]