What is the value of `[((tan5theta+tan3theta))/(4cos4theta(tan5theta-tan3theta))]`?

Options

A. `sin2theta`
B. `cos2theta`
C. `tan4theta`
D. `cot2theta`

Answer

To find the value of the expression \(\frac{\tan 5\theta + \tan 3\theta}{4\cos 4\theta (\tan 5\theta - \tan 3\theta)}\), we can follow these steps:

### Step 1: Rewrite the Tangent Functions
We start by expressing the tangent functions in terms of sine and cosine:
\[ \tan 5\theta = \frac{\sin 5\theta}{\cos 5\theta} \quad \text{and} \quad \tan 3\theta = \frac{\sin 3\theta}{\cos 3\theta} \]
Thus, we can rewrite the expression as:
\[ \frac{\frac{\sin 5\theta}{\cos 5\theta} + \frac{\sin 3\theta}{\cos 3\theta}}{4\cos 4\theta \left(\frac{\sin 5\theta}{\cos 5\theta} - \frac{\sin 3\theta}{\cos 3\theta}\right)} \]

### Step 2: Combine the Numerator
The numerator becomes:
\[ \frac{\sin 5\theta \cos 3\theta + \sin 3\theta \cos 5\theta}{\cos 5\theta \cos 3\theta} \]

### Step 3: Combine the Denominator
The denominator becomes:
\[ 4\cos 4\theta \cdot \frac{\sin 5\theta \cos 3\theta - \sin 3\theta \cos 5\theta}{\cos 5\theta \cos 3\theta} \]

### Step 4: Simplify the Expression
Now we can simplify the entire expression:
\[ \frac{\sin 5\theta \cos 3\theta + \sin 3\theta \cos 5\theta}{\sin 5\theta \cos 3\theta - \sin 3\theta \cos 5\theta} \cdot \frac{1}{4\cos 4\theta} \]
Using the sine addition and subtraction formulas:
\[ \sin(A + B) = \sin A \cos B + \cos A \sin B \quad \text{and} \quad \sin(A - B) = \sin A \cos B - \cos A \sin B \]
we can rewrite the expression as:
\[ \frac{\sin(5\theta + 3\theta)}{\sin(5\theta - 3\theta)} \cdot \frac{1}{4\cos 4\theta} \]
This simplifies to:
\[ \frac{\sin 8\theta}{\sin 2\theta} \cdot \frac{1}{4\cos 4\theta} \]

### Step 5: Further Simplification
Using the identity \(\sin 2\theta = 2\sin \theta \cos \theta\), we can rewrite:
\[ \frac{\sin 8\theta}{2\sin \theta \cos \theta} \cdot \frac{1}{4\cos 4\theta} \]
This gives us:
\[ \frac{\sin 8\theta}{8\sin \theta \cos \theta \cos 4\theta} \]

### Step 6: Final Result
After simplifying, we find that the expression can be evaluated based on the values of \(\theta\). In particular, if we consider specific values for \(\theta\), we can find the final numerical value.