The greatest value of `sin^4theta+cos^4theta` is

Options

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

Explanation

To find the greatest value of \( \sin^4 \theta + \cos^4 \theta \), we can follow these steps:

### Step 1: Use the identity for \( \sin^4 \theta + \cos^4 \theta \)

We can rewrite \( \sin^4 \theta + \cos^4 \theta \) using the identity:
\[ \sin^4 \theta + \cos^4 \theta = (\sin^2 \theta + \cos^2 \theta)^2 - 2 \sin^2 \theta \cos^2 \theta \]
Since \( \sin^2 \theta + \cos^2 \theta = 1 \), we can substitute this into our equation:
\[ \sin^4 \theta + \cos^4 \theta = 1^2 - 2 \sin^2 \theta \cos^2 \theta = 1 - 2 \sin^2 \theta \cos^2 \theta \]

### Step 2: Simplify \( \sin^2 \theta \cos^2 \theta \)

Next, we can express \( \sin^2 \theta \cos^2 \theta \) in terms of \( \sin^2 \theta \):
\[ \sin^2 \theta \cos^2 \theta = \frac{1}{4} \sin^2(2\theta) \]
This is because \( \sin(2\theta) = 2 \sin \theta \cos \theta \).

### Step 3: Substitute back into the equation

Now substituting this back into our equation gives:
\[ \sin^4 \theta + \cos^4 \theta = 1 - 2 \cdot \frac{1}{4} \sin^2(2\theta) = 1 - \frac{1}{2} \sin^2(2\theta) \]

### Step 4: Find the maximum value

The term \( \sin^2(2\theta) \) varies between 0 and 1. Therefore, the maximum value of \( 1 - \frac{1}{2} \sin^2(2\theta) \) occurs when \( \sin^2(2\theta) \) is at its minimum value, which is 0:
\[ \text{Maximum value} = 1 - \frac{1}{2} \cdot 0 = 1 \]

### Conclusion

Thus, the greatest value of \( \sin^4 \theta + \cos^4 \theta \) is:
\[ \boxed{1} \]