If `sin^(4)A-cos^(4)A=1,then (A//2)`is ______.`(0ltAle90^(@))`.

Options

A. `45^(@)`
B. `60^(@)`
C. `30^(@)`
D. `40^(@)`

Explanation

To solve the equation \( \sin^4 A - \cos^4 A = 1 \), we can follow these steps:

### Step 1: Recognize the difference of squares
The expression \( \sin^4 A - \cos^4 A \) can be factored using the difference of squares formula:
\[ \sin^4 A - \cos^4 A = (\sin^2 A + \cos^2 A)(\sin^2 A - \cos^2 A) \]
Since \( \sin^2 A + \cos^2 A = 1 \), we can simplify the equation to:
\[ 1(\sin^2 A - \cos^2 A) = 1 \]
Thus, we have:
\[ \sin^2 A - \cos^2 A = 1 \]

### Step 2: Rewrite \( \sin^2 A \)
From the equation \( \sin^2 A - \cos^2 A = 1 \), we can express \( \sin^2 A \) in terms of \( \cos^2 A \):
\[ \sin^2 A = \cos^2 A + 1 \]

### Step 3: Use the Pythagorean identity
Using the identity \( \sin^2 A + \cos^2 A = 1 \), we can substitute for \( \sin^2 A \):
\[ \cos^2 A + 1 + \cos^2 A = 1 \]
This simplifies to:
\[ 2\cos^2 A + 1 = 1 \]

### Step 4: Solve for \( \cos^2 A \)
Rearranging gives:
\[ 2\cos^2 A = 0 \]
Thus:
\[ \cos^2 A = 0 \]

### Step 5: Find \( A \)
Since \( \cos^2 A = 0 \), we have:
\[ \cos A = 0 \]
The angle \( A \) that satisfies this in the range \( 0 < A < 90^\circ \) is:
\[ A = 90^\circ \]

### Step 6: Calculate \( \frac{A}{2} \)
Now, we find \( \frac{A}{2} \):
\[ \frac{A}{2} = \frac{90^\circ}{2} = 45^\circ \]

### Final Answer
Thus, the value of \( \frac{A}{2} \) is \( 45^\circ \).

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