If `5 costheta = 3`, then `( "cosec "theta+cot theta)/(" cosec " theta- cot theta)` is equal to

Answer

To solve the problem, we start with the given equation and the expression we need to evaluate.

**Step 1: Find cos(θ)**
Given that \( 5 \cos \theta = 3 \), we can isolate \( \cos \theta \):
\[ \cos \theta = \frac{3}{5} \]

**Step 2: Rewrite the expression**
We need to evaluate the expression:
\[ \frac{\csc \theta + \cot \theta}{\csc \theta - \cot \theta} \]
Recall that:
\[ \csc \theta = \frac{1}{\sin \theta} \quad \text{and} \quad \cot \theta = \frac{\cos \theta}{\sin \theta} \]
Thus, we can rewrite the expression as:
\[ \frac{\frac{1}{\sin \theta} + \frac{\cos \theta}{\sin \theta}}{\frac{1}{\sin \theta} - \frac{\cos \theta}{\sin \theta}} \]

**Step 3: Simplify the expression**
Factoring out \( \frac{1}{\sin \theta} \) from both the numerator and the denominator gives:
\[ \frac{\frac{1 + \cos \theta}{\sin \theta}}{\frac{1 - \cos \theta}{\sin \theta}} = \frac{1 + \cos \theta}{1 - \cos \theta} \]

**Step 4: Substitute the value of cos(θ)**
Now, substitute \( \cos \theta = \frac{3}{5} \):
\[ \frac{1 + \frac{3}{5}}{1 - \frac{3}{5}} = \frac{\frac{5}{5} + \frac{3}{5}}{\frac{5}{5} - \frac{3}{5}} = \frac{\frac{8}{5}}{\frac{2}{5}} \]

**Step 5: Simplify the fraction**
Now simplify the fraction:
\[ \frac{\frac{8}{5}}{\frac{2}{5}} = \frac{8}{5} \times \frac{5}{2} = \frac{8 \cdot 5}{5 \cdot 2} = \frac{8}{2} = 4 \]

Thus, the final answer is:
\[ \boxed{4} \]

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