If `5 cos theta = 3` , evaluate : `("cosec" theta-cottheta)/("cosec" theta + cottheta)`

Answer

To solve the problem, we will follow these steps:

### Step 1: Find the value of cos θ
Given the equation:
\[ 5 \cos \theta = 3 \]
To find \( \cos \theta \), we divide both sides by 5:
\[ \cos \theta = \frac{3}{5} \]

### Step 2: Rewrite the expression
We need to evaluate:
\[ \frac{\csc \theta - \cot \theta}{\csc \theta + \cot \theta} \]
We know that:
- \( \csc \theta = \frac{1}{\sin \theta} \)
- \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

Substituting these into the expression gives us:
\[ \frac{\frac{1}{\sin \theta} - \frac{\cos \theta}{\sin \theta}}{\frac{1}{\sin \theta} + \frac{\cos \theta}{\sin \theta}} \]

### Step 3: Simplify the expression
We can factor out \( \frac{1}{\sin \theta} \) from both the numerator and the denominator:
\[ = \frac{\frac{1 - \cos \theta}{\sin \theta}}{\frac{1 + \cos \theta}{\sin \theta}} \]

The \( \sin \theta \) cancels out:
\[ = \frac{1 - \cos \theta}{1 + \cos \theta} \]

### Step 4: Substitute the value of cos θ
Now we substitute \( \cos \theta = \frac{3}{5} \) into the expression:
\[ = \frac{1 - \frac{3}{5}}{1 + \frac{3}{5}} \]

### Step 5: Simplify the numerator and denominator
Calculating the numerator:
\[ 1 - \frac{3}{5} = \frac{5}{5} - \frac{3}{5} = \frac{2}{5} \]

Calculating the denominator:
\[ 1 + \frac{3}{5} = \frac{5}{5} + \frac{3}{5} = \frac{8}{5} \]

### Step 6: Final simplification
Now we have:
\[ = \frac{\frac{2}{5}}{\frac{8}{5}} \]

This simplifies to:
\[ = \frac{2}{5} \times \frac{5}{8} = \frac{2}{8} = \frac{1}{4} \]

### Final Answer
Thus, the final answer is:
\[ \frac{1}{4} \]

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