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

Options

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

Answer

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

### Step 1: Rewrite the Expression
We start with the expression:
\[ \sin^4 \theta + \cos^4 \theta \]
We can use the identity for the sum of fourth powers:
\[ a^4 + b^4 = (a^2 + b^2)^2 - 2a^2b^2 \]
Let \( a = \sin^2 \theta \) and \( b = \cos^2 \theta \). Thus, we can rewrite the expression as:
\[ \sin^4 \theta + \cos^4 \theta = (\sin^2 \theta + \cos^2 \theta)^2 - 2\sin^2 \theta \cos^2 \theta \]

### Step 2: Use the Pythagorean Identity
We know that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting this into our expression gives:
\[ \sin^4 \theta + \cos^4 \theta = 1^2 - 2\sin^2 \theta \cos^2 \theta = 1 - 2\sin^2 \theta \cos^2 \theta \]

### Step 3: Express in Terms of Sine Double Angle
Next, we can express \( \sin^2 \theta \cos^2 \theta \) using the double angle identity:
\[ \sin^2 \theta \cos^2 \theta = \left(\frac{1}{2} \sin 2\theta\right)^2 = \frac{1}{4} \sin^2 2\theta \]
Substituting this back into our expression 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 4: Find the Maximum Value
To find the maximum value of \( \sin^4 \theta + \cos^4 \theta \), we need to minimize \( \sin^2 2\theta \). The minimum value of \( \sin^2 2\theta \) is 0 (since the sine function oscillates between -1 and 1). Thus:
\[ \text{Minimum of } \sin^2 2\theta = 0 \]
Substituting this back into our expression gives:
\[ \sin^4 \theta + \cos^4 \theta = 1 - \frac{1}{2} \cdot 0 = 1 \]

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