What is the value of `[ ( cos 7 A + cos 5 A) div ( sin 7 a - sin 5 A)]` ?

Options

A. tan A
B. tan 4 A
C. cot 4 A
D. cot A

Answer

To solve the problem \(\frac{\cos 7A + \cos 5A}{\sin 7A - \sin 5A}\), we can use trigonometric identities to simplify the expression.

### Step-by-Step Solution:

1. **Use the Cosine Addition Identity**:
The identity for the sum of cosines is:
\[ \cos a + \cos b = 2 \cos\left(\frac{a+b}{2}\right) \cos\left(\frac{a-b}{2}\right) \]
Here, let \(a = 7A\) and \(b = 5A\).
\[ \cos 7A + \cos 5A = 2 \cos\left(\frac{7A + 5A}{2}\right) \cos\left(\frac{7A - 5A}{2}\right) = 2 \cos(6A) \cos(A) \]

2. **Use the Sine Subtraction Identity**:
The identity for the difference of sines is:
\[ \sin a - \sin b = 2 \sin\left(\frac{a-b}{2}\right) \cos\left(\frac{a+b}{2}\right) \]
Again, using \(a = 7A\) and \(b = 5A\):
\[ \sin 7A - \sin 5A = 2 \sin\left(\frac{7A - 5A}{2}\right) \cos\left(\frac{7A + 5A}{2}\right) = 2 \sin(A) \cos(6A) \]

3. **Substituting Back into the Original Expression**:
Now substitute the results from steps 1 and 2 into the original expression:
\[ \frac{\cos 7A + \cos 5A}{\sin 7A - \sin 5A} = \frac{2 \cos(6A) \cos(A)}{2 \sin(A) \cos(6A)} \]

4. **Canceling Common Terms**:
The \(2\) and \(\cos(6A)\) in the numerator and denominator can be canceled (assuming \(\cos(6A) \neq 0\)):
\[ = \frac{\cos(A)}{\sin(A)} \]

5. **Final Result**:
The expression simplifies to:
\[ \frac{\cos(A)}{\sin(A)} = \cot(A) \]

### Final Answer:
The value of \(\frac{\cos 7A + \cos 5A}{\sin 7A - \sin 5A}\) is \(\cot(A)\).