`2 ( sin^(6)A + cos^(6) A) - 3 ( sin^(4) A + cos^(4) A )` is

Options

A. 2
B. 0
C. `-1`
D. none

Answer

To solve the expression \(2 (\sin^6 A + \cos^6 A) - 3 (\sin^4 A + \cos^4 A)\), we will simplify it step by step.

### Step 1: Rewrite the expression
We start with the expression:
\[ 2 (\sin^6 A + \cos^6 A) - 3 (\sin^4 A + \cos^4 A) \]

### Step 2: Use the identity for \(a^3 + b^3\)
Recall that \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\). We can apply this to \(\sin^6 A + \cos^6 A\):
\[ \sin^6 A + \cos^6 A = (\sin^2 A)^3 + (\cos^2 A)^3 = (\sin^2 A + \cos^2 A)((\sin^2 A)^2 - \sin^2 A \cos^2 A + (\cos^2 A)^2) \]
Since \(\sin^2 A + \cos^2 A = 1\), we have:
\[ \sin^6 A + \cos^6 A = 1 \cdot ((\sin^2 A)^2 - \sin^2 A \cos^2 A + (\cos^2 A)^2) \]
Now, \((\sin^2 A)^2 + (\cos^2 A)^2 = \sin^4 A + \cos^4 A\), so:
\[ \sin^6 A + \cos^6 A = \sin^4 A + \cos^4 A - \sin^2 A \cos^2 A \]

### Step 3: Substitute back into the expression
Substituting this back into our original expression gives:
\[ 2(\sin^4 A + \cos^4 A - \sin^2 A \cos^2 A) - 3(\sin^4 A + \cos^4 A) \]

### Step 4: Distribute the terms
Distributing the 2:
\[ 2\sin^4 A + 2\cos^4 A - 2\sin^2 A \cos^2 A - 3\sin^4 A - 3\cos^4 A \]

### Step 5: Combine like terms
Now, combine the terms:
\[ (2\sin^4 A - 3\sin^4 A) + (2\cos^4 A - 3\cos^4 A) - 2\sin^2 A \cos^2 A \]
This simplifies to:
\[ -\sin^4 A - \cos^4 A - 2\sin^2 A \cos^2 A \]

### Step 6: Use the identity for \(\sin^4 A + \cos^4 A\)
Recall that:
\[ \sin^4 A + \cos^4 A = (\sin^2 A + \cos^2 A)^2 - 2\sin^2 A \cos^2 A = 1 - 2\sin^2 A \cos^2 A \]
Substituting this into our expression gives:
\[ -(1 - 2\sin^2 A \cos^2 A) - 2\sin^2 A \cos^2 A \]
This simplifies to:
\[ -1 + 2\sin^2 A \cos^2 A - 2\sin^2 A \cos^2 A = -1 \]

### Final Answer
Thus, the final answer is:
\[ \boxed{-1} \]