If `A=sin^(2)theta+cos^(4)theta`, for any value of `theta` , then the value of A is

Options

A. `1leAle2`
B. `(3)/(4)leAle1`
C. `(13)/(16)leAle1`
D. `(3)/(4)leAle(13)/(16)`

Answer

To solve the problem, we need to find the value of \( A \) given by the expression:

\[ A = \sin^2 \theta + \cos^4 \theta \]

### Step 1: Rewrite the expression for \( A \)

We start with the expression for \( A \):

\[ A = \sin^2 \theta + \cos^4 \theta \]

### Step 2: Express \( \cos^4 \theta \) in terms of \( \sin^2 \theta \)

Using the identity \( \cos^2 \theta = 1 - \sin^2 \theta \), we can express \( \cos^4 \theta \) as:

\[ \cos^4 \theta = (\cos^2 \theta)^2 = (1 - \sin^2 \theta)^2 \]

### Step 3: Expand \( \cos^4 \theta \)

Now, we expand \( (1 - \sin^2 \theta)^2 \):

\[ \cos^4 \theta = 1 - 2\sin^2 \theta + \sin^4 \theta \]

### Step 4: Substitute back into the expression for \( A \)

Substituting this back into the expression for \( A \):

\[ A = \sin^2 \theta + (1 - 2\sin^2 \theta + \sin^4 \theta) \]

### Step 5: Combine like terms

Combining the terms, we get:

\[ A = 1 - \sin^2 \theta + \sin^4 \theta \]

### Step 6: Let \( x = \sin^2 \theta \)

Let \( x = \sin^2 \theta \). Then, we can rewrite \( A \) as:

\[ A = 1 - x + x^2 \]

### Step 7: Analyze the quadratic expression

The expression \( A = x^2 - x + 1 \) is a quadratic function in terms of \( x \). The vertex of a quadratic \( ax^2 + bx + c \) is given by \( x = -\frac{b}{2a} \). Here, \( a = 1 \) and \( b = -1 \):

\[ x = -\frac{-1}{2 \cdot 1} = \frac{1}{2} \]

### Step 8: Find the maximum and minimum values of \( A \)

Now, we evaluate \( A \) at \( x = 0 \) and \( x = 1 \) (the boundaries of \( \sin^2 \theta \)):

- For \( x = 0 \):

\[ A = 1 - 0 + 0^2 = 1 \]

- For \( x = 1 \):

\[ A = 1 - 1 + 1^2 = 1 \]

- For \( x = \frac{1}{2} \):

\[ A = \left(\frac{1}{2}\right)^2 - \left(\frac{1}{2}\right) + 1 = \frac{1}{4} - \frac{1}{2} + 1 = \frac{3}{4} \]

### Conclusion

Thus, the minimum value of \( A \) is \( \frac{3}{4} \) and the maximum value is \( 1 \). Therefore, the range of \( A \) is:

\[ \frac{3}{4} \leq A \leq 1 \]

### Final Answer

The value of \( A \) is in the range \( \frac{3}{4} \leq A \leq 1 \).

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