The expression
`(cos 6x+6cos 4x+15 cos 2x+10)/(cos 5x+5 cos 3x+10 cos x)`

Options

A. cos 2x
B. 2 cos x
C. `cos^(2)x`
D. 1+cos x

Answer

To simplify the expression

\[ \frac{\cos 6x + 6\cos 4x + 15\cos 2x + 10}{\cos 5x + 5\cos 3x + 10\cos x}, \]

we will follow these steps:

### Step 1: Simplify the Numerator
The numerator is

\[ \cos 6x + 6\cos 4x + 15\cos 2x + 10. \]

We can rewrite \(15\cos 2x\) as \(5\cos 2x + 10\cos 2x\) and \(10\) as \(10\cos 0\) (since \(\cos 0 = 1\)). Thus, we can express the numerator as:

\[ \cos 6x + 6\cos 4x + 5\cos 2x + 10\cos 2x + 10\cos 0. \]

### Step 2: Rearranging the Terms
Now we can rearrange the terms in the numerator:

\[ \cos 6x + 6\cos 4x + 5\cos 2x + 10\cos 2x + 10\cos 0 = \cos 6x + 6\cos 4x + 15\cos 2x + 10. \]

### Step 3: Factor Out Common Terms
Next, we can factor out common terms from the numerator. Notice that we can group \(6\cos 4x\) and \(15\cos 2x\):

\[ \cos 6x + 6\cos 4x + 15\cos 2x + 10 = \cos 6x + 6(\cos 4x + \frac{15}{6}\cos 2x + \frac{10}{6}). \]

### Step 4: Simplify the Denominator
Now, let's simplify the denominator:

\[ \cos 5x + 5\cos 3x + 10\cos x. \]

We can factor out \(5\) from the second term:

\[ \cos 5x + 5(\cos 3x + 2\cos x). \]

### Step 5: Use Trigonometric Identities
We can apply the sum-to-product identities. For example, we can use:

\[ \cos A + \cos B = 2\cos\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right). \]

### Step 6: Final Simplification
After applying the identities and simplifying, we will find that the numerator and denominator share common terms, allowing us to cancel them out.

### Final Result
After simplification, we find that the expression simplifies to:

\[ 2\cos x. \]