What is the value of `[(tan 5 theta + tan 3theta)//4 cos 4theta (tan 5theta - tan 3theta)]`?

Options

A. `sin 2 theta`
B. `cos 2theta`
C. `tan 4theta`
D. `cot 2theta`

Answer

To solve the expression \(\frac{\tan 5\theta + \tan 3\theta}{4 \cos 4\theta (\tan 5\theta - \tan 3\theta)}\), we will follow these steps:

### Step 1: Rewrite the Tangent Functions
Using the definition of tangent, we can express \(\tan\) in terms of sine and cosine:
\[ \tan 5\theta = \frac{\sin 5\theta}{\cos 5\theta}, \quad \tan 3\theta = \frac{\sin 3\theta}{\cos 3\theta} \]

### Step 2: Substitute into the Expression
Substituting these definitions into our expression gives:
\[ \frac{\frac{\sin 5\theta}{\cos 5\theta} + \frac{\sin 3\theta}{\cos 3\theta}}{4 \cos 4\theta \left(\frac{\sin 5\theta}{\cos 5\theta} - \frac{\sin 3\theta}{\cos 3\theta}\right)} \]

### Step 3: Combine the Numerator
The numerator can be combined as follows:
\[ \frac{\sin 5\theta \cos 3\theta + \sin 3\theta \cos 5\theta}{\cos 5\theta \cos 3\theta} \]
This uses the identity for the sum of tangents.

### Step 4: Combine the Denominator
The denominator can be simplified:
\[ \frac{\sin 5\theta \cos 3\theta - \sin 3\theta \cos 5\theta}{\cos 5\theta \cos 3\theta} \]
This uses the identity for the difference of tangents.

### Step 5: Substitute Back into the Expression
Now substituting back into our expression gives:
\[ \frac{\sin 5\theta \cos 3\theta + \sin 3\theta \cos 5\theta}{\cos 5\theta \cos 3\theta} \cdot \frac{\cos 5\theta \cos 3\theta}{4 \cos 4\theta (\sin 5\theta \cos 3\theta - \sin 3\theta \cos 5\theta)} \]

### Step 6: Simplify the Expression
The \(\cos 5\theta \cos 3\theta\) cancels out:
\[ \frac{\sin 5\theta \cos 3\theta + \sin 3\theta \cos 5\theta}{4 \cos 4\theta (\sin 5\theta \cos 3\theta - \sin 3\theta \cos 5\theta)} \]

### Step 7: Use Sine Addition and Subtraction Formulas
Using the sine addition and subtraction formulas:
\[ \sin(5\theta + 3\theta) = \sin 8\theta, \quad \sin(5\theta - 3\theta) = \sin 2\theta \]
Thus, we can rewrite the expression as:
\[ \frac{\sin 8\theta}{4 \cos 4\theta \sin 2\theta} \]

### Step 8: Apply the Double Angle Formula
Using the double angle formula for sine:
\[ \sin 8\theta = 2 \sin 4\theta \cos 4\theta \]
Substituting this in gives:
\[ \frac{2 \sin 4\theta \cos 4\theta}{4 \cos 4\theta \sin 2\theta} \]

### Step 9: Cancel Common Terms
Canceling \(\cos 4\theta\) from numerator and denominator:
\[ \frac{2 \sin 4\theta}{4 \sin 2\theta} = \frac{\sin 4\theta}{2 \sin 2\theta} \]

### Step 10: Use the Double Angle Formula Again
Using the double angle formula again:
\[ \sin 4\theta = 2 \sin 2\theta \cos 2\theta \]
Substituting this gives:
\[ \frac{2 \sin 2\theta \cos 2\theta}{2 \sin 2\theta} \]

### Step 11: Final Simplification
Canceling \(2 \sin 2\theta\) gives:
\[ \cos 2\theta \]

### Final Answer
Thus, the value of the expression is:
\[ \cos 2\theta \]