If `A = sin^2 theta+ cos^4 theta`, then for all real values of `theta`

Answer

To solve the problem where \( A = \sin^2 \theta + \cos^4 \theta \), we will analyze and simplify the expression step by step.

### Step 1: Rewrite the expression
We start with the expression:
\[ A = \sin^2 \theta + \cos^4 \theta \]
We can express \( \cos^4 \theta \) in terms of \( \sin^2 \theta \):
\[ \cos^4 \theta = (\cos^2 \theta)^2 = (1 - \sin^2 \theta)^2 \]
Thus, we can rewrite \( A \) as:
\[ A = \sin^2 \theta + (1 - \sin^2 \theta)^2 \]

### Step 2: Expand the squared term
Now, we expand \( (1 - \sin^2 \theta)^2 \):
\[ (1 - \sin^2 \theta)^2 = 1 - 2\sin^2 \theta + \sin^4 \theta \]
Substituting this back into the expression for \( A \):
\[ A = \sin^2 \theta + 1 - 2\sin^2 \theta + \sin^4 \theta \]
This simplifies to:
\[ A = 1 - \sin^2 \theta + \sin^4 \theta \]

### Step 3: Let \( x = \sin^2 \theta \)
Let \( x = \sin^2 \theta \). Then, \( A \) can be rewritten as:
\[ A = 1 - x + x^2 \]
where \( x \) ranges from \( 0 \) to \( 1 \) (since \( \sin^2 \theta \) can take values between 0 and 1).

### Step 4: Find the maximum and minimum values of \( A \)
To find the maximum and minimum values of \( A \), we can analyze the quadratic function:
\[ A = x^2 - x + 1 \]
This is a parabola that opens upwards. The vertex of the parabola gives us the minimum value. The vertex \( x \) can be found using the formula \( x = -\frac{b}{2a} \):
\[ x = -\frac{-1}{2 \cdot 1} = \frac{1}{2} \]

### Step 5: Calculate \( A \) at the vertex
Now, we substitute \( x = \frac{1}{2} \) back into the expression for \( A \):
\[ A = \left(\frac{1}{2}\right)^2 - \frac{1}{2} + 1 = \frac{1}{4} - \frac{2}{4} + \frac{4}{4} = \frac{3}{4} \]

### Step 6: Evaluate \( A \) at the endpoints
Next, we evaluate \( A \) at the endpoints of the interval \( x = 0 \) and \( x = 1 \):
- For \( x = 0 \):
\[ A = 1 - 0 + 0^2 = 1 \]
- For \( x = 1 \):
\[ A = 1 - 1 + 1^2 = 1 \]

### Conclusion
Thus, the minimum value of \( A \) is \( \frac{3}{4} \) and the maximum value is \( 1 \). Therefore, for all real values of \( \theta \):
\[ \frac{3}{4} \leq A \leq 1 \]