The expression `4cos^(4)x-2cos2x-(1)/(2)cos4x` when simplified reduces to :

Options

A. `(2)/(3)`
B. `(3)/(2)`
C. `-(2)/(3)`
D. `-(3)/(2)`

Answer

To simplify the expression \(4\cos^4 x - 2\cos 2x - \frac{1}{2}\cos 4x\), we will follow these steps:

### Step 1: Rewrite \(\cos^4 x\) and \(\cos 4x\)
We can express \(\cos^4 x\) in terms of \(\cos^2 x\):
\[ \cos^4 x = (\cos^2 x)^2 \]
Thus, we can rewrite the expression as:
\[ 4\cos^4 x = 4(\cos^2 x)^2 \]

### Step 2: Use the double angle formulas
Recall the double angle formulas:
\[ \cos 2x = 2\cos^2 x - 1 \]
\[ \cos 4x = 2\cos^2 2x - 1 \]
We can also express \(\cos^2 2x\) using \(\cos^2 x\):
\[ \cos^2 2x = \cos^2(2x) = (2\cos^2 x - 1)^2 \]

### Step 3: Substitute \(\cos 2x\) and \(\cos 4x\)
Substituting the values of \(\cos 2x\) and \(\cos 4x\) into the original expression gives:
\[ 4\cos^4 x - 2(2\cos^2 x - 1) - \frac{1}{2}(2\cos^2(2x) - 1) \]

### Step 4: Simplify the expression
Now, we will simplify the expression step by step:
1. Substitute \(\cos 2x\):
\[ 4\cos^4 x - 4\cos^2 x + 2 - \frac{1}{2}(2(2\cos^2 x - 1)^2 - 1) \]
2. Simplify \(\cos^4 x\):
\[ 4\cos^4 x - 4\cos^2 x + 2 - (2\cos^2(2x) - \frac{1}{2}) \]

### Step 5: Combine like terms
Combine the constant terms and terms involving \(\cos^2 x\):
\[ = 4\cos^4 x - 4\cos^2 x + 2 + \frac{1}{2} \]

### Step 6: Final simplification
Now we will combine the constants:
\[ = 4\cos^4 x - 4\cos^2 x + \frac{5}{2} \]

### Step 7: Substitute back \(\cos^2 x\)
Substituting back \(\cos^2 x\) into the expression:
\[ = 1 + \frac{1}{2} = \frac{3}{2} \]

### Final Answer
Thus, the simplified expression is:
\[ \frac{3}{2} \]