Find the value of:
`2("sin"^(6)x+cos^(6)x)-3("sin"^(4)x+cos^(4)x)+2`.

Answer

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

### Step 1: Use the identity for \( \sin^6 x + \cos^6 x \)

We can use the identity:
\[ a^3 + b^3 = (a + b)(a^2 - ab + b^2) \]
where \( a = \sin^2 x \) and \( b = \cos^2 x \).

Thus,
\[ \sin^6 x + \cos^6 x = (\sin^2 x + \cos^2 x)((\sin^2 x)^2 - \sin^2 x \cos^2 x + (\cos^2 x)^2) \]

Since \( \sin^2 x + \cos^2 x = 1 \), we have:
\[ \sin^6 x + \cos^6 x = 1 \cdot ((\sin^2 x)^2 - \sin^2 x \cos^2 x + (\cos^2 x)^2) \]

Now, we can simplify \( (\sin^2 x)^2 + (\cos^2 x)^2 \):
\[ (\sin^2 x)^2 + (\cos^2 x)^2 = \sin^4 x + \cos^4 x \]

So,
\[ \sin^6 x + \cos^6 x = \sin^4 x + \cos^4 x - \sin^2 x \cos^2 x \]

### Step 2: Substitute back into the original expression

Now substitute this back into the original expression:
\[ 2(\sin^6 x + \cos^6 x) = 2(\sin^4 x + \cos^4 x - \sin^2 x \cos^2 x) \]
\[ = 2\sin^4 x + 2\cos^4 x - 2\sin^2 x \cos^2 x \]

Now, substituting this into the expression:
\[ 2\sin^4 x + 2\cos^4 x - 2\sin^2 x \cos^2 x - 3(\sin^4 x + \cos^4 x) + 2 \]

### Step 3: Combine like terms

Now combine the terms:
\[ (2\sin^4 x - 3\sin^4 x) + (2\cos^4 x - 3\cos^4 x) - 2\sin^2 x \cos^2 x + 2 \]
\[ = -\sin^4 x - \cos^4 x - 2\sin^2 x \cos^2 x + 2 \]

### Step 4: Use the identity for \( \sin^4 x + \cos^4 x \)

We can use the identity:
\[ \sin^4 x + \cos^4 x = (\sin^2 x + \cos^2 x)^2 - 2\sin^2 x \cos^2 x = 1 - 2\sin^2 x \cos^2 x \]

Substituting this back:
\[ -\left(1 - 2\sin^2 x \cos^2 x\right) - 2\sin^2 x \cos^2 x + 2 \]
\[ = -1 + 2\sin^2 x \cos^2 x - 2\sin^2 x \cos^2 x + 2 \]
\[ = -1 + 2 \]
\[ = 1 \]

### Final Result

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