If `10 sec^(2)x - 9tan^(2)x= 13 and 0^@le x lt 90^@` then x = _____

Options

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

Answer

To solve the equation \(10 \sec^2 x - 9 \tan^2 x = 13\) for \(0^\circ \leq x < 90^\circ\), we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ 10 \sec^2 x - 9 \tan^2 x = 13 \]

### Step 2: Use the identity
We know the trigonometric identity:
\[ \sec^2 x = 1 + \tan^2 x \]
Using this identity, we can express \(\sec^2 x\) in terms of \(\tan^2 x\):
\[ \sec^2 x = 1 + \tan^2 x \]
Substituting this into the equation gives:
\[ 10(1 + \tan^2 x) - 9 \tan^2 x = 13 \]

### Step 3: Simplify the equation
Expanding the equation:
\[ 10 + 10 \tan^2 x - 9 \tan^2 x = 13 \]
This simplifies to:
\[ 10 + \tan^2 x = 13 \]

### Step 4: Isolate \(\tan^2 x\)
Now, we isolate \(\tan^2 x\):
\[ \tan^2 x = 13 - 10 \]
\[ \tan^2 x = 3 \]

### Step 5: Find \(\tan x\)
Taking the square root of both sides, we find:
\[ \tan x = \sqrt{3} \]

### Step 6: Determine \(x\)
We know that:
\[ \tan 60^\circ = \sqrt{3} \]
Thus, we conclude:
\[ x = 60^\circ \]

### Final Answer
Therefore, the value of \(x\) is:
\[ \boxed{60^\circ} \]