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 squares
We start with the expression \( \sin^4 \theta + \cos^4 \theta \). We can rewrite this using the identity \( a^2 + b^2 = (a + b)^2 - 2ab \).

Let \( a = \sin^2 \theta \) and \( b = \cos^2 \theta \). Then, we have:
\[ \sin^4 \theta + \cos^4 \theta = a^2 + b^2 = (a + b)^2 - 2ab \]

### Step 2: Substitute the identity for sine and cosine
Since \( \sin^2 \theta + \cos^2 \theta = 1 \), we can substitute:
\[ (a + b)^2 = 1^2 = 1 \]
Thus, we have:
\[ \sin^4 \theta + \cos^4 \theta = 1 - 2ab \]

### Step 3: Express \( ab \) in terms of sine and cosine
Now, we need to express \( ab \):
\[ ab = \sin^2 \theta \cos^2 \theta = \frac{1}{4} \sin^2(2\theta) \]
This is because \( \sin(2\theta) = 2 \sin \theta \cos \theta \).

### Step 4: Substitute \( ab \) back into the equation
Substituting \( ab \) into our equation gives:
\[ \sin^4 \theta + \cos^4 \theta = 1 - 2 \left(\frac{1}{4} \sin^2(2\theta)\right) = 1 - \frac{1}{2} \sin^2(2\theta) \]

### Step 5: Find the maximum value
The maximum value of \( \sin^2(2\theta) \) is 1 (since the sine function ranges from -1 to 1). Therefore, the minimum value of \( \frac{1}{2} \sin^2(2\theta) \) is 0.

So, the greatest value of \( \sin^4 \theta + \cos^4 \theta \) occurs when \( \sin^2(2\theta) = 0 \):
\[ \sin^4 \theta + \cos^4 \theta = 1 - 0 = 1 \]

### Conclusion
Thus, the greatest value of \( \sin^4 \theta + \cos^4 \theta \) is \( \boxed{1} \).

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