If `4cos^(2)x^(@)-1=0 and 0 le x^(@) le 90^(@)`, find :
`1/(cos^(2)x^(@))-tan^(2)x^(@)`

Answer

To solve the equation \(4 \cos^2 x - 1 = 0\) and find the value of \(\frac{1}{\cos^2 x} - \tan^2 x\), follow these steps:

### Step 1: Solve the equation
Starting with the equation:
\[ 4 \cos^2 x - 1 = 0 \]
Add 1 to both sides:
\[ 4 \cos^2 x = 1 \]
Now, divide both sides by 4:
\[ \cos^2 x = \frac{1}{4} \]

### Step 2: Take the square root
Taking the square root of both sides gives:
\[ \cos x = \pm \frac{1}{2} \]
Since \(0 \leq x \leq 90^\circ\), we only consider the positive value:
\[ \cos x = \frac{1}{2} \]

### Step 3: Find the angle
The angle \(x\) that satisfies \(\cos x = \frac{1}{2}\) is:
\[ x = 60^\circ \]

### Step 4: Calculate \(\frac{1}{\cos^2 x} - \tan^2 x\)
Now we need to calculate:
\[ \frac{1}{\cos^2 x} - \tan^2 x \]
First, find \(\cos^2 60^\circ\):
\[ \cos^2 60^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]
Then, calculate \(\tan 60^\circ\):
\[ \tan 60^\circ = \sqrt{3} \]
Now, find \(\tan^2 60^\circ\):
\[ \tan^2 60^\circ = (\sqrt{3})^2 = 3 \]

### Step 5: Substitute values into the expression
Now substitute the values into the expression:
\[ \frac{1}{\cos^2 60^\circ} - \tan^2 60^\circ = \frac{1}{\frac{1}{4}} - 3 \]
This simplifies to:
\[ 4 - 3 = 1 \]

### Final Answer
Thus, the value of \(\frac{1}{\cos^2 x} - \tan^2 x\) is:
\[ \boxed{1} \]