$\frac{cos~7x+cos~5x}{sin~7x-sin~5x}=$

Q: $\frac{cos~7x+cos~5x}{sin~7x-sin~5x}=$

To solve the given expression $\frac{\cos 7x + \cos 5x}{\sin 7x - \sin 5x}$, we can use trigonometric identities to simplify it and verify why Option 1, $\cot x$, is the correct answer. ### Step-by-step Explanation: 1. **Use Sum-to-Product Identities:** The sum-to-product identities for cosine and sine are: \[ \cos A + \cos B = 2 \cos \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right) \] \[ \sin A - \sin B = 2 \cos \left(\frac{A+B}{2}\right) \sin \left(\frac{A-B}{2}\right) \] 2. **Apply the Identities:** For the numerator $\cos 7x + \cos 5x$: \[ \cos 7x + \cos 5x = 2 \cos \left(\frac{7x + 5x}{2}\right) \cos \left(\frac{7x - 5x}{2}\right) = 2 \cos (6x) \cos (x) \] For the denominator $\sin 7x - \sin 5x$: \[ \sin 7x - \sin 5x = 2 \cos \left(\frac{7x + 5x}{2}\right) \sin \left(\frac{7x - 5x}{2}\right) = 2 \cos (6x) \sin (x) \] 3. **Simplify the Expression:** Substitute these into the original expression: \[ \frac{\cos 7x + \cos 5x}{\sin 7x - \sin 5x} = \frac{2 \cos (6x) \cos (x)}{2 \cos (6x) \sin (x)} \] Cancel out the common factor $2 \cos (6x)$: \[ = \frac{\cos (x)}{\sin (x)} = \cot x \] Thus, the expression simplifies to $\cot x$, which confirms that Option 1 is correct. ### Explanation of Incorrect Options: - **Option 2: $\tan x$** - The expression simplifies to $\cot x$, not $\tan x$. $\tan x$ is the reciprocal of $\cot x$, so this option is incorrect. - **Option 3: $\sin x$** - The expression involves a ratio of cosine to sine, which simplifies to $\cot x$, not $\sin x$. Therefore, this option is incorrect. - **Option 4: $\cot 7x$** - The simplification process does not yield $\cot 7x$. The expression specifically simplifies to $\cot x$, making this option incorrect. In conclusion, the correct answer is indeed Option 1, $\cot x$, as derived from the simplification using trigonometric identities.

Question

Accepted Answer

To solve the given expression $\frac{\cos 7x + \cos 5x}{\sin 7x - \sin 5x}$, we can use trigonometric identities to simplify it and verify why Option 1, $\cot x$, is the correct answer.

### Step-by-step Explanation:

1. **Use Sum-to-Product Identities:**

The sum-to-product identities for cosine and sine are:
   $$
   \cos A + \cos B = 2 \cos \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)
   $$
   $$
   \sin A - \sin B = 2 \cos \left(\frac{A+B}{2}\right) \sin \left(\frac{A-B}{2}\right)
   $$

2. **Apply the Identities:**

For the numerator $\cos 7x + \cos 5x$:
   $$
   \cos 7x + \cos 5x = 2 \cos \left(\frac{7x + 5x}{2}\right) \cos \left(\frac{7x - 5x}{2}\right) = 2 \cos (6x) \cos (x)
   $$

For the denominator $\sin 7x - \sin 5x$:
   $$
   \sin 7x - \sin 5x = 2 \cos \left(\frac{7x + 5x}{2}\right) \sin \left(\frac{7x - 5x}{2}\right) = 2 \cos (6x) \sin (x)
   $$

3. **Simplify the Expression:**

Substitute these into the original expression:
   $$
   \frac{\cos 7x + \cos 5x}{\sin 7x - \sin 5x} = \frac{2 \cos (6x) \cos (x)}{2 \cos (6x) \sin (x)}
   $$

Cancel out the common factor $2 \cos (6x)$:
   $$
   = \frac{\cos (x)}{\sin (x)} = \cot x
   $$

Thus, the expression simplifies to $\cot x$, which confirms that Option 1 is correct.

### Explanation of Incorrect Options:

- **Option 2: $\tan x$**
  - The expression simplifies to $\cot x$, not $\tan x$. $\tan x$ is the reciprocal of $\cot x$, so this option is incorrect.

- **Option 3: $\sin x$**
  - The expression involves a ratio of cosine to sine, which simplifies to $\cot x$, not $\sin x$. Therefore, this option is incorrect.

- **Option 4: $\cot 7x$**
  - The simplification process does not yield $\cot 7x$. The expression specifically simplifies to $\cot x$, making this option incorrect.

In conclusion, the correct answer is indeed Option 1, $\cot x$, as derived from the simplification using trigonometric identities.