The expression `(cos6x+6cos4x+15cos2x+10)/(cos5x+5cos3x+10cosx)` is equal to

Options

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

Explanation

To solve the expression

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

we will simplify both the numerator and the denominator step by step.

### Step 1: Simplifying the Numerator

The numerator is:

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

We can rearrange and group the terms:

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

Now, we can rewrite \(10\) as \(10\cos 0\):

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

### Step 2: Using Cosine Addition Formula

Next, we can use the cosine addition formula to combine terms. The formula states:

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

We can apply this to \(\cos 6x + \cos 4x\):

\[ \cos 6x + \cos 4x = 2 \cos \left(\frac{6x + 4x}{2}\right) \cos \left(\frac{6x - 4x}{2}\right) = 2 \cos 5x \cos x \]

So, we can rewrite the numerator as:

\[ = 2 \cos 5x \cos x + 5\cos 2x + 10\cos 2x + 10 \]

This simplifies to:

\[ = 2 \cos 5x \cos x + 15\cos 2x + 10 \]

### Step 3: Simplifying the Denominator

The denominator is:

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

We can group the terms similarly:

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

### Step 4: Factor Out Common Terms

Now, we can factor out common terms from both the numerator and denominator.

From the numerator:

\[ = 2\cos x (\cos 5x + 7.5\cos 2x + 5) \]

From the denominator:

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

### Step 5: Final Simplification

Now, we can simplify the entire expression:

\[ \frac{2\cos x (\cos 5x + 7.5\cos 2x + 5)}{\cos 5x + 5\cos 3x + 10\cos x} \]

If we assume that \(\cos 5x + 5\cos 3x + 10\cos x\) is not zero, we can cancel out \(\cos x\):

\[ = 2 \]

### Final Answer

Thus, the expression simplifies to:

\[ \boxed{2} \]