`cos^(2)A(3-4cos^(2)A)^(2)+sin^(2)A(3-4sin^(2)A)^(2)=?`

Options

A. 1
B. sin4A
C. cos4A
D. 0

Answer

To solve the expression \( \cos^2 A (3 - 4 \cos^2 A)^2 + \sin^2 A (3 - 4 \sin^2 A)^2 \), we can follow these steps:

### Step 1: Identify the terms
We have two main components in the expression:
1. \( \cos^2 A (3 - 4 \cos^2 A)^2 \)
2. \( \sin^2 A (3 - 4 \sin^2 A)^2 \)

### Step 2: Rewrite the expression
We can rewrite the expression as:
\[ \cos^2 A \cdot (3 - 4 \cos^2 A)^2 + \sin^2 A \cdot (3 - 4 \sin^2 A)^2 \]

### Step 3: Expand the squares
Now, we will expand both squares:
1. For \( (3 - 4 \cos^2 A)^2 \):
\[ (3 - 4 \cos^2 A)^2 = 9 - 24 \cos^2 A + 16 \cos^4 A \]
2. For \( (3 - 4 \sin^2 A)^2 \):
\[ (3 - 4 \sin^2 A)^2 = 9 - 24 \sin^2 A + 16 \sin^4 A \]

### Step 4: Substitute back into the expression
Substituting these expansions back into the original expression gives:
\[ \cos^2 A (9 - 24 \cos^2 A + 16 \cos^4 A) + \sin^2 A (9 - 24 \sin^2 A + 16 \sin^4 A) \]

### Step 5: Distribute \( \cos^2 A \) and \( \sin^2 A \)
Now, distribute \( \cos^2 A \) and \( \sin^2 A \):
\[ 9 \cos^2 A - 24 \cos^4 A + 16 \cos^6 A + 9 \sin^2 A - 24 \sin^4 A + 16 \sin^6 A \]

### Step 6: Combine like terms
Combine the terms involving \( \cos^2 A \) and \( \sin^2 A \):
\[ 9 (\cos^2 A + \sin^2 A) - 24 (\cos^4 A + \sin^4 A) + 16 (\cos^6 A + \sin^6 A) \]

### Step 7: Use the Pythagorean identity
Using the identity \( \cos^2 A + \sin^2 A = 1 \):
\[ 9 \cdot 1 - 24 (\cos^4 A + \sin^4 A) + 16 (\cos^6 A + \sin^6 A) \]

### Step 8: Simplify further
Now we need to simplify \( \cos^4 A + \sin^4 A \) and \( \cos^6 A + \sin^6 A \):
1. \( \cos^4 A + \sin^4 A = (\cos^2 A + \sin^2 A)^2 - 2 \cos^2 A \sin^2 A = 1 - 2 \cos^2 A \sin^2 A \)
2. \( \cos^6 A + \sin^6 A = (\cos^2 A + \sin^2 A)(\cos^4 A + \sin^4 A - \cos^2 A \sin^2 A) = 1 \cdot \left( (1 - 2 \cos^2 A \sin^2 A) - \cos^2 A \sin^2 A \right) = 1 - 3 \cos^2 A \sin^2 A \)

### Step 9: Substitute back
Substituting these back:
\[ 9 - 24(1 - 2 \cos^2 A \sin^2 A) + 16(1 - 3 \cos^2 A \sin^2 A) \]
\[ = 9 - 24 + 48 \cos^2 A \sin^2 A + 16 - 48 \cos^2 A \sin^2 A \]
\[ = 1 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{1} \]