If `cosectheta+ 3 sec theta= 5cosectheta` then what is the value of `cot theta` ?

Options

A. `4//3`
B. `3//4`
C. `1//sqrt(3)`
D. `sqrt(3)`

Answer

To solve the equation \( \csc \theta + 3 \sec \theta = 5 \csc \theta \) and find the value of \( \cot \theta \), we can follow these steps:

### Step 1: Rewrite the equation using sine and cosine
We know that:
- \( \csc \theta = \frac{1}{\sin \theta} \)
- \( \sec \theta = \frac{1}{\cos \theta} \)

Substituting these into the equation gives:
\[ \frac{1}{\sin \theta} + 3 \cdot \frac{1}{\cos \theta} = 5 \cdot \frac{1}{\sin \theta} \]

### Step 2: Find a common denominator
The common denominator for the left-hand side is \( \sin \theta \cos \theta \). Thus, we can rewrite the equation as:
\[ \frac{\cos \theta + 3 \sin \theta}{\sin \theta \cos \theta} = \frac{5}{\sin \theta} \]

### Step 3: Cross-multiply to eliminate the fractions
Cross-multiplying gives:
\[ \cos \theta + 3 \sin \theta = 5 \cos \theta \]

### Step 4: Rearrange the equation
Rearranging this equation leads to:
\[ 3 \sin \theta = 5 \cos \theta - \cos \theta \]
\[ 3 \sin \theta = 4 \cos \theta \]

### Step 5: Solve for \( \tan \theta \)
Dividing both sides by \( \cos \theta \) gives:
\[ \tan \theta = \frac{4}{3} \]

### Step 6: Find \( \cot \theta \)
Since \( \cot \theta = \frac{1}{\tan \theta} \), we have:
\[ \cot \theta = \frac{1}{\frac{4}{3}} = \frac{3}{4} \]

### Final Answer
Thus, the value of \( \cot \theta \) is \( \frac{3}{4} \).

---