Certainly! Let's solve the problem step-by-step.
### Step 1: Given Information and Initial Calculations
We are given that [tex]\(\tan \theta = \frac{1}{\sqrt{5}}\)[/tex].
### Step 2: Finding [tex]\(\sec^2 \theta\)[/tex]
We start by recalling the trigonometric identity:
[tex]\[
\sec^2 \theta = 1 + \tan^2 \theta
\][/tex]
Since [tex]\(\tan \theta = \frac{1}{\sqrt{5}}\)[/tex], we find:
[tex]\[
\tan^2 \theta = \left(\frac{1}{\sqrt{5}}\right)^2 = \frac{1}{5}
\][/tex]
Next, we compute [tex]\(\sec^2 \theta\)[/tex]:
[tex]\[
\sec^2 \theta = 1 + \frac{1}{5} = \frac{5}{5} + \frac{1}{5} = \frac{6}{5} = 1.2
\][/tex]
### Step 3: Finding [tex]\(\operatorname{cosec}^2 \theta\)[/tex]
We recall another trigonometric identity:
[tex]\[
\operatorname{cosec}^2 \theta = 1 + \cot^2 \theta
\][/tex]
Using the fact that:
[tex]\[
\cot \theta = \frac{1}{\tan \theta} = \sqrt{5}
\][/tex]
Then, we compute [tex]\(\cot^2 \theta\)[/tex]:
[tex]\[
\cot^2 \theta = (\sqrt{5})^2 = 5
\][/tex]
So, [tex]\(\operatorname{cosec}^2 \theta\)[/tex] is:
[tex]\[
\operatorname{cosec}^2 \theta = 1 + 5 = 6.0
\][/tex]
### Step 4: Calculating the Numerator and Denominator
We need to calculate the expression:
[tex]\[
\frac{\operatorname{cosec}^2 \theta - \sec^2 \theta}{\operatorname{cosec}^2 \theta + \sec^2 \theta}
\][/tex]
First, find the numerator:
[tex]\[
\operatorname{cosec}^2 \theta - \sec^2 \theta = 6.0 - 1.2 = 4.8
\][/tex]
Next, find the denominator:
[tex]\[
\operatorname{cosec}^2 \theta + \sec^2 \theta = 6.0 + 1.2 = 7.2
\][/tex]
### Step 5: Finding the Final Expression
Finally, compute the value of the expression:
[tex]\[
\frac{\operatorname{cosec}^2 \theta - \sec^2 \theta}{\operatorname{cosec}^2 \theta + \sec^2 \theta} = \frac{4.8}{7.2} = \frac{2}{3} = 0.6666666666666666
\][/tex]
### Summary
The value of [tex]\(\frac{\operatorname{cosec}^2 \theta - \sec^2 \theta}{\operatorname{cosec}^2 \theta + \sec^2 \theta}\)[/tex] is approximately [tex]\(0.6666666666666666\)[/tex], or [tex]\(\frac{2}{3}\)[/tex].
