If `3sec^(2)x - 2tan^(2)x = 6` and `0^(@) le x le 90^(@)` then x = ?

Options

A. `60^(@)`
B. `30^(@)`
C. `45^(@)`
D. `90^(@)`

Answer

To solve the equation \(3 \sec^2 x - 2 \tan^2 x = 6\) for \(0^\circ \leq x \leq 90^\circ\), we can follow these steps:

### Step 1: Use the identity for \(\sec^2 x\)
We know that:
\[ \sec^2 x = 1 + \tan^2 x \]
Substituting this identity into the equation gives us:
\[ 3(1 + \tan^2 x) - 2 \tan^2 x = 6 \]

### Step 2: Simplify the equation
Expanding the equation:
\[ 3 + 3 \tan^2 x - 2 \tan^2 x = 6 \]
This simplifies to:
\[ 3 + \tan^2 x = 6 \]

### Step 3: Isolate \(\tan^2 x\)
Now, subtract 3 from both sides:
\[ \tan^2 x = 6 - 3 \]
\[ \tan^2 x = 3 \]

### Step 4: Take the square root
Taking the square root of both sides gives us:
\[ \tan x = \sqrt{3} \quad \text{or} \quad \tan x = -\sqrt{3} \]
Since \(0^\circ \leq x \leq 90^\circ\), we only consider the positive value:
\[ \tan x = \sqrt{3} \]

### Step 5: Find the angle \(x\)
The angle \(x\) for which \(\tan x = \sqrt{3}\) is:
\[ x = 60^\circ \]

### Conclusion
Thus, the solution to the equation \(3 \sec^2 x - 2 \tan^2 x = 6\) in the interval \(0^\circ \leq x \leq 90^\circ\) is:
\[ \boxed{60^\circ} \]