If `sec A=sqrt2`, find : `(3 cot^2A+2sin^2A)/(tan^2A-cos^2A)`

Answer

To solve the problem step by step, we need to find the value of the expression \((3 \cot^2 A + 2 \sin^2 A) / (tan^2 A - \cos^2 A)\) given that \(\sec A = \sqrt{2}\).

### Step 1: Find \(\tan^2 A\)
We know that:
\[ \sec^2 A = 1 + \tan^2 A \]
Substituting \(\sec A = \sqrt{2}\):
\[ \sec^2 A = (\sqrt{2})^2 = 2 \]
Thus:
\[ 2 = 1 + \tan^2 A \implies \tan^2 A = 2 - 1 = 1 \]

### Step 2: Find \(\cot^2 A\)
Using the relationship between cotangent and tangent:
\[ \cot^2 A = \frac{1}{\tan^2 A} \]
Since we found \(\tan^2 A = 1\):
\[ \cot^2 A = \frac{1}{1} = 1 \]

### Step 3: Find \(\cos^2 A\)
Using the relationship between secant and cosine:
\[ \cos^2 A = \frac{1}{\sec^2 A} \]
Substituting \(\sec^2 A = 2\):
\[ \cos^2 A = \frac{1}{2} \]

### Step 4: Find \(\sin^2 A\)
Using the Pythagorean identity:
\[ \sin^2 A + \cos^2 A = 1 \]
Substituting \(\cos^2 A = \frac{1}{2}\):
\[ \sin^2 A = 1 - \cos^2 A = 1 - \frac{1}{2} = \frac{1}{2} \]

### Step 5: Substitute values into the expression
Now we substitute the values we found into the expression:
\[ \frac{3 \cot^2 A + 2 \sin^2 A}{\tan^2 A - \cos^2 A} \]
Substituting \(\cot^2 A = 1\), \(\sin^2 A = \frac{1}{2}\), \(\tan^2 A = 1\), and \(\cos^2 A = \frac{1}{2}\):
\[ = \frac{3(1) + 2\left(\frac{1}{2}\right)}{1 - \frac{1}{2}} \]
Calculating the numerator:
\[ = \frac{3 + 1}{1 - \frac{1}{2}} = \frac{4}{\frac{1}{2}} = 4 \times 2 = 8 \]

### Final Answer
The value of the expression is \(8\).

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