`3(sin x- cos x )^(4) + 6(sin x+ cos x )^(2) +4 (sin ^(6) x+ cos ^(6) x)=`

Options

A. 14
B. 11
C. 12
D. 13

Answer

To solve the expression \( 3(\sin x - \cos x)^4 + 6(\sin x + \cos x)^2 + 4(\sin^6 x + \cos^6 x) \), we can follow these steps:

### Step 1: Expand \( (\sin x - \cos x)^4 \)
Using the binomial theorem:
\[ (\sin x - \cos x)^4 = \sin^4 x - 4\sin^3 x \cos x + 6\sin^2 x \cos^2 x - 4\sin x \cos^3 x + \cos^4 x \]

### Step 2: Expand \( (\sin x + \cos x)^2 \)
Using the formula for the square of a sum:
\[ (\sin x + \cos x)^2 = \sin^2 x + 2\sin x \cos x + \cos^2 x = 1 + 2\sin x \cos x \]

### Step 3: Expand \( \sin^6 x + \cos^6 x \)
Using the identity \( a^6 + b^6 = (a^2 + b^2)(a^4 - a^2b^2 + b^4) \):
\[ \sin^6 x + \cos^6 x = (\sin^2 x + \cos^2 x)((\sin^2 x)^2 - \sin^2 x \cos^2 x + (\cos^2 x)^2) \]
Since \( \sin^2 x + \cos^2 x = 1 \):
\[ = \sin^4 x - \sin^2 x \cos^2 x + \cos^4 x \]

### Step 4: Substitute the expansions into the original expression
Now substituting back into the original expression:
\[ 3(\sin^4 x - 4\sin^3 x \cos x + 6\sin^2 x \cos^2 x - 4\sin x \cos^3 x + \cos^4 x) + 6(1 + 2\sin x \cos x) + 4(\sin^4 x - \sin^2 x \cos^2 x + \cos^4 x) \]

### Step 5: Combine like terms
Combine all terms:
- Collect terms involving \( \sin^4 x \) and \( \cos^4 x \)
- Collect terms involving \( \sin^2 x \cos^2 x \)
- Collect terms involving \( \sin^3 x \cos x \) and \( \sin x \cos^3 x \)
- Collect constant terms

### Step 6: Simplify the expression
After combining and simplifying, you will arrive at a final expression which can be further simplified if necessary.

### Final Result
The final result of the expression will yield a simplified form based on the calculations above.