If `8 sec^(2)x- 7 tan^(2)x= 11 and 0^(@) le x le 90^(@)`, then x= ?

Options

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

Answer

To solve the equation \( 8 \sec^2 x - 7 \tan^2 x = 11 \) for \( 0^\circ \leq x \leq 90^\circ \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ 8 \sec^2 x - 7 \tan^2 x = 11 \]

### Step 2: Use the identity
Recall the trigonometric identity:
\[ \sec^2 x = 1 + \tan^2 x \]
Substituting this identity into the equation gives:
\[ 8(1 + \tan^2 x) - 7 \tan^2 x = 11 \]

### Step 3: Simplify the equation
Expanding the equation:
\[ 8 + 8 \tan^2 x - 7 \tan^2 x = 11 \]
This simplifies to:
\[ 8 + \tan^2 x = 11 \]

### Step 4: Isolate \(\tan^2 x\)
Subtract 8 from both sides:
\[ \tan^2 x = 11 - 8 \]
Thus:
\[ \tan^2 x = 3 \]

### Step 5: Solve for \(\tan x\)
Taking the square root of both sides:
\[ \tan x = \sqrt{3} \]

### Step 6: Find the angle \(x\)
The angle \(x\) for which \(\tan x = \sqrt{3}\) in the range \(0^\circ \leq x \leq 90^\circ\) is:
\[ x = 60^\circ \]

### Final Answer
Thus, the solution to the equation is:
\[ \boxed{60^\circ} \]