What is `(sin5x-sin3x)/(cos5x+cos3x)` equal to?

Options

A. sin x
B. cos x
C. tan x
D. cot x

Answer

To solve the expression \((\sin 5x - \sin 3x) / (\cos 5x + \cos 3x)\), we will use trigonometric identities.

### Step-by-Step Solution:

1. **Apply the Sine Difference Formula**:
We use the formula for the difference of sines:
\[ \sin A - \sin B = 2 \cos\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right) \]
Here, let \(A = 5x\) and \(B = 3x\). Thus, we have:
\[ \sin 5x - \sin 3x = 2 \cos\left(\frac{5x + 3x}{2}\right) \sin\left(\frac{5x - 3x}{2}\right) \]
Simplifying this gives:
\[ \sin 5x - \sin 3x = 2 \cos(4x) \sin(x) \]

2. **Apply the Cosine Sum Formula**:
We use the formula for the sum of cosines:
\[ \cos A + \cos B = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \]
Again, let \(A = 5x\) and \(B = 3x\). Thus, we have:
\[ \cos 5x + \cos 3x = 2 \cos\left(\frac{5x + 3x}{2}\right) \cos\left(\frac{5x - 3x}{2}\right) \]
Simplifying this gives:
\[ \cos 5x + \cos 3x = 2 \cos(4x) \cos(x) \]

3. **Substituting Back into the Original Expression**:
Now we substitute these results back into the original expression:
\[ \frac{\sin 5x - \sin 3x}{\cos 5x + \cos 3x} = \frac{2 \cos(4x) \sin(x)}{2 \cos(4x) \cos(x)} \]

4. **Canceling Common Terms**:
The \(2 \cos(4x)\) in the numerator and denominator cancels out (assuming \(\cos(4x) \neq 0\)):
\[ = \frac{\sin(x)}{\cos(x)} = \tan(x) \]

### Final Answer:
Thus, the expression \((\sin 5x - \sin 3x) / (\cos 5x + \cos 3x)\) simplifies to:
\[ \tan(x) \]