Step 1: Understanding the Concept:
We are given the value of $\sin\theta$ and asked to evaluate a complex trigonometric expression. We can either find the values of all trigonometric functions and substitute them, or simplify the expression first.
Step 2: Key Formula or Approach:
From $5 \sin\theta = 4$, we get $\sin\theta = \frac{4}{5}$.
We can find $\cos\theta$ using $\cos^2\theta = 1 - \sin^2\theta$. Then find $\csc\theta = 1/\sin\theta$ and $\cot\theta = \cos\theta/\sin\theta$.
Alternatively, simplify the expression:
\[ \frac{\csc\theta - \cot\theta}{\csc\theta + \cot\theta} = \frac{\frac{1}{\sin\theta} - \frac{\cos\theta}{\sin\theta}}{\frac{1}{\sin\theta} + \frac{\cos\theta}{\sin\theta}} = \frac{\frac{1-\cos\theta}{\sin\theta}}{\frac{1+\cos\theta}{\sin\theta}} = \frac{1-\cos\theta}{1+\cos\theta} \]
Step 3: Detailed Explanation:
Given $\sin\theta = \frac{4}{5}$. Since the options are all positive, we can assume $\theta$ is in the first quadrant, where all trigonometric ratios are positive.
First, find $\cos\theta$:
\[ \cos^2\theta = 1 - \sin^2\theta = 1 - \left(\frac{4}{5}\right)^2 = 1 - \frac{16}{25} = \frac{9}{25} \]
\[ \cos\theta = \sqrt{\frac{9}{25}} = \frac{3}{5} \quad \text{(since we assume Q1)} \]
Now, use the simplified expression from Step 2:
\[ \frac{1-\cos\theta}{1+\cos\theta} = \frac{1 - \frac{3}{5}}{1 + \frac{3}{5}} \]
\[ = \frac{\frac{5-3}{5}}{\frac{5+3}{5}} = \frac{\frac{2}{5}}{\frac{8}{5}} \]
\[ = \frac{2}{8} = \frac{1}{4} \]
If we had not assumed Q1, $\cos\theta$ could also be $-\frac{3}{5}$ (for Q2). In that case, the expression would be $\frac{1 - (-3/5)}{1 + (-3/5)} = \frac{8/5}{2/5} = 4$. Since 4 is not an option, the question implies the acute angle case.
Step 4: Final Answer:
The value of the expression is $\frac{1}{4}$. Therefore, option (D) is the correct answer.