Certainly! Let's break this problem down methodically, step-by-step.
### Problem Breakdown
We are given:
[tex]\[
\operatorname{cosec} A = \sqrt{2}
\][/tex]
We need to find the value of the following expression:
[tex]\[
\frac{2 \sin^2 A + 3 \cot^2 A}{\tan^2 A - \cos^2 A}
\][/tex]
### Step-by-Step Solution
1. Calculate [tex]\(\sin A\)[/tex]:
Given [tex]\(\operatorname{cosec} A = \sqrt{2}\)[/tex], we recall the relationship between cosecant and sine:
[tex]\[
\operatorname{cosec} A = \frac{1}{\sin A}
\][/tex]
Thus,
[tex]\[
\sin A = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}
\][/tex]
2. Calculate [tex]\(\sin^2 A\)[/tex]:
[tex]\[
\sin^2 A = \left(\frac{\sqrt{2}}{2}\right)^2 = \frac{2}{4} = \frac{1}{2}
\][/tex]
3. Calculate [tex]\(\cos^2 A\)[/tex]:
Using the Pythagorean identity [tex]\(\sin^2 A + \cos^2 A = 1\)[/tex], we find [tex]\(\cos^2 A\)[/tex]:
[tex]\[
\cos^2 A = 1 - \sin^2 A = 1 - \frac{1}{2} = \frac{1}{2}
\][/tex]
4. Calculate [tex]\(\cot^2 A\)[/tex]:
Recall that [tex]\(\cot A = \frac{\cos A}{\sin A}\)[/tex], thus:
[tex]\[
\cot^2 A = \left(\frac{\cos A}{\sin A}\right)^2 = \frac{\cos^2 A}{\sin^2 A} = \frac{\frac{1}{2}}{\frac{1}{2}} = 1
\][/tex]
5. Calculate [tex]\(\tan^2 A\)[/tex]:
Recall that [tex]\(\tan A = \frac{\sin A}{\cos A}\)[/tex], thus:
[tex]\[
\tan^2 A = \left(\frac{\sin A}{\cos A}\right)^2 = \frac{\sin^2 A}{\cos^2 A} = \frac{\frac{1}{2}}{\frac{1}{2}} = 1
\][/tex]
6. Calculate the numerator: [tex]\(2 \sin^2 A + 3 \cot^2 A\)[/tex]
Substitute the values we calculated:
[tex]\[
2 \sin^2 A + 3 \cot^2 A = 2 \left(\frac{1}{2}\right) + 3(1) = 1 + 3 = 4
\][/tex]
7. Calculate the denominator: [tex]\(\tan^2 A - \cos^2 A\)[/tex]
Substitute the values we calculated:
[tex]\[
\tan^2 A - \cos^2 A = 1 - \frac{1}{2} = \frac{1}{2}
\][/tex]
8. Compute the final expression:
Substitute the values of the numerator and the denominator:
[tex]\[
\frac{2 \sin^2 A + 3 \cot^2 A}{\tan^2 A - \cos^2 A} = \frac{4}{\frac{1}{2}} = 4 \times 2 = 8
\][/tex]
### Final Answer:
[tex]\[
\boxed{8}
\][/tex]
