Find maximum & minimum value of `sin^(6)theta+cos^(6)theta`

Answer

To find the maximum and minimum values of the expression \( \sin^6 \theta + \cos^6 \theta \), we can follow these steps:

### Step 1: Recognize the expression
We start with the expression:
\[ y = \sin^6 \theta + \cos^6 \theta \]

### Step 2: Use the identity for powers
We can use the identity \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \) where \( a = \sin^2 \theta \) and \( b = \cos^2 \theta \). Thus, we can rewrite:
\[ y = (\sin^2 \theta)^3 + (\cos^2 \theta)^3 = (u^3 + v^3) \]
where \( u = \sin^2 \theta \) and \( v = \cos^2 \theta \).

### Step 3: Apply the identity
Using the identity:
\[ y = (u + v)(u^2 - uv + v^2) \]
Since \( u + v = \sin^2 \theta + \cos^2 \theta = 1 \), we have:
\[ y = 1 \cdot (u^2 - uv + v^2) = u^2 - uv + v^2 \]

### Step 4: Substitute \( u \) and \( v \)
Now, substituting \( u = \sin^2 \theta \) and \( v = \cos^2 \theta \):
\[ y = \sin^4 \theta + \cos^4 \theta - \sin^2 \theta \cos^2 \theta \]

### Step 5: Use the identity for squares
We can further simplify \( \sin^4 \theta + \cos^4 \theta \) using the identity:
\[ \sin^4 \theta + \cos^4 \theta = (\sin^2 \theta + \cos^2 \theta)^2 - 2\sin^2 \theta \cos^2 \theta = 1 - 2\sin^2 \theta \cos^2 \theta \]
Thus:
\[ y = 1 - 2\sin^2 \theta \cos^2 \theta - \sin^2 \theta \cos^2 \theta = 1 - 3\sin^2 \theta \cos^2 \theta \]

### Step 6: Substitute \( \sin^2 \theta \cos^2 \theta \)
Let \( z = \sin^2 \theta \cos^2 \theta \). The maximum value of \( z \) occurs when \( \sin^2 \theta = \cos^2 \theta = \frac{1}{2} \), giving:
\[ z = \frac{1}{4} \]
Thus:
\[ y = 1 - 3z = 1 - 3 \cdot \frac{1}{4} = 1 - \frac{3}{4} = \frac{1}{4} \]

### Step 7: Determine maximum and minimum values
From our calculations:
- The maximum value of \( y \) occurs when either \( \sin^2 \theta = 1 \) or \( \cos^2 \theta = 1 \), which gives \( y = 1 \).
- The minimum value occurs when \( \sin^2 \theta = \cos^2 \theta = \frac{1}{2} \), which gives \( y = \frac{1}{4} \).

### Final Result
Thus, the maximum value of \( \sin^6 \theta + \cos^6 \theta \) is \( 1 \) and the minimum value is \( \frac{1}{4} \).