The expression `(cos6x+6cos4x+15cos2x+10)/(cos5x+5cos3x+10cosx) = lambdacosx`, then the value of `lambda` is ___________.

Answer

To solve the expression

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

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 use the cosine addition formula and identities to simplify this. We will rewrite \(6\cos 4x\) and \(15\cos 2x\):

- \(6\cos 4x = 2(\cos 4x + \cos 4x + \cos 4x)\)
- \(15\cos 2x = 5(\cos 2x + \cos 2x + \cos 2x + \cos 2x + \cos 2x)\)

Now, we can group terms:

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