What is `(cos7x-cos3x)/(sin7x-2sin5x+sin3x)`equal to ?

Options

A. tan x
B. cot x
C. tan 2x
D. cot 2x

Explanation

To solve the expression \((\cos 7x - \cos 3x) / (\sin 7x - 2\sin 5x + \sin 3x)\), we will use trigonometric identities to simplify it step by step.

### Step 1: Apply the Cosine Difference Identity
We start with the numerator \(\cos 7x - \cos 3x\). We can use the cosine difference identity:
\[ \cos A - \cos B = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \]
Here, \(A = 7x\) and \(B = 3x\). Thus:
\[ \cos 7x - \cos 3x = -2 \sin\left(\frac{7x + 3x}{2}\right) \sin\left(\frac{7x - 3x}{2}\right) = -2 \sin(5x) \sin(2x) \]

### Step 2: Simplify the Denominator
Now, we simplify the denominator \(\sin 7x - 2\sin 5x + \sin 3x\). We can group \(\sin 7x\) and \(\sin 3x\):
\[ \sin 7x + \sin 3x = 2 \sin\left(\frac{7x + 3x}{2}\right) \cos\left(\frac{7x - 3x}{2}\right) = 2 \sin(5x) \cos(2x) \]
Thus, the denominator becomes:
\[ \sin 7x - 2\sin 5x + \sin 3x = 2 \sin(5x) \cos(2x) - 2\sin(5x) = 2\sin(5x)(\cos(2x) - 1) \]

### Step 3: Substitute Back into the Expression
Now we can substitute our results back into the original expression:
\[ \frac{\cos 7x - \cos 3x}{\sin 7x - 2\sin 5x + \sin 3x} = \frac{-2 \sin(5x) \sin(2x)}{2\sin(5x)(\cos(2x) - 1)} \]

### Step 4: Cancel Common Terms
We can cancel \(2\sin(5x)\) from the numerator and denominator (assuming \(\sin(5x) \neq 0\)):
\[ = \frac{-\sin(2x)}{\cos(2x) - 1} \]

### Step 5: Simplify Further
Using the identity \(1 - \cos(2x) = 2\sin^2(x)\), we can rewrite the denominator:
\[ = \frac{-\sin(2x)}{-2\sin^2(x)} = \frac{\sin(2x)}{2\sin^2(x)} \]

### Step 6: Final Simplification
Using the double angle formula \(\sin(2x) = 2\sin(x)\cos(x)\), we can simplify further:
\[ = \frac{2\sin(x)\cos(x)}{2\sin^2(x)} = \frac{\cos(x)}{\sin(x)} = \cot(x) \]

### Final Answer
Thus, the expression \(\frac{\cos 7x - \cos 3x}{\sin 7x - 2\sin 5x + \sin 3x}\) simplifies to:
\[ \cot(x) \]