If ` tan theta =(1)/(sqrt(5)), " write the value of " (("cosec"^(2)theta-sec^(2)theta))/(("cosec"^(2)theta+sec^(2)theta)).`

Explanation

To solve the problem, we start with the given information:

1. **Given**: \( \tan \theta = \frac{1}{\sqrt{5}} \)

We need to find the value of:

\[ \frac{\csc^2 \theta - \sec^2 \theta}{\csc^2 \theta + \sec^2 \theta} \]

### Step 1: Finding \( \csc^2 \theta \) and \( \sec^2 \theta \)

Using the identity \( \csc^2 \theta = 1 + \cot^2 \theta \) and \( \sec^2 \theta = 1 + \tan^2 \theta \):

- First, we calculate \( \tan^2 \theta \):
\[ \tan^2 \theta = \left( \frac{1}{\sqrt{5}} \right)^2 = \frac{1}{5} \]

- Now, we find \( \sec^2 \theta \):
\[ \sec^2 \theta = 1 + \tan^2 \theta = 1 + \frac{1}{5} = \frac{5}{5} + \frac{1}{5} = \frac{6}{5} \]

- Next, we find \( \cot^2 \theta \):
\[ \cot \theta = \frac{1}{\tan \theta} = \sqrt{5} \quad \Rightarrow \quad \cot^2 \theta = 5 \]

- Now, we find \( \csc^2 \theta \):
\[ \csc^2 \theta = 1 + \cot^2 \theta = 1 + 5 = 6 \]

### Step 2: Substituting \( \csc^2 \theta \) and \( \sec^2 \theta \) into the expression

Now we substitute \( \csc^2 \theta \) and \( \sec^2 \theta \) into the expression:

\[ \frac{\csc^2 \theta - \sec^2 \theta}{\csc^2 \theta + \sec^2 \theta} = \frac{6 - \frac{6}{5}}{6 + \frac{6}{5}} \]

### Step 3: Simplifying the expression

- First, simplify the numerator:
\[ 6 - \frac{6}{5} = \frac{30}{5} - \frac{6}{5} = \frac{24}{5} \]

- Now, simplify the denominator:
\[ 6 + \frac{6}{5} = \frac{30}{5} + \frac{6}{5} = \frac{36}{5} \]

### Step 4: Final computation

Now we can substitute these back into the expression:

\[ \frac{\frac{24}{5}}{\frac{36}{5}} = \frac{24}{36} = \frac{2}{3} \]

Thus, the final answer is:

\[ \frac{\csc^2 \theta - \sec^2 \theta}{\csc^2 \theta + \sec^2 \theta} = \frac{2}{3} \]