If ` 0 le theta le (pi)/(2)` and `sec^(2) theta + tan^(2) theta = 7` , then `theta` is

Options

A. `(15pi)/(12)` radian
B. `(pi)/(3)` radian
C. `(pi)/(5)` radian
D. `(pi)/(6)` radian

Answer

To solve the equation \( \sec^2 \theta + \tan^2 \theta = 7 \) given that \( 0 \leq \theta \leq \frac{\pi}{2} \), we can follow these steps:

### Step 1: Recall the identity
We know from trigonometric identities that:
\[ \sec^2 \theta - \tan^2 \theta = 1 \]
Let’s denote this as Equation (1).

### Step 2: Set up the equations
We have two equations now:
1. \( \sec^2 \theta + \tan^2 \theta = 7 \) (Equation 2)
2. \( \sec^2 \theta - \tan^2 \theta = 1 \) (Equation 1)

### Step 3: Add the two equations
Adding Equation (1) and Equation (2):
\[ (\sec^2 \theta + \tan^2 \theta) + (\sec^2 \theta - \tan^2 \theta) = 7 + 1 \]
This simplifies to:
\[ 2\sec^2 \theta = 8 \]

### Step 4: Solve for \( \sec^2 \theta \)
Dividing both sides by 2:
\[ \sec^2 \theta = 4 \]

### Step 5: Find \( \sec \theta \)
Taking the square root:
\[ \sec \theta = 2 \]

### Step 6: Find \( \theta \)
Since \( \sec \theta = \frac{1}{\cos \theta} \), we have:
\[ \cos \theta = \frac{1}{2} \]
The angle \( \theta \) that satisfies \( \cos \theta = \frac{1}{2} \) in the range \( 0 \leq \theta \leq \frac{\pi}{2} \) is:
\[ \theta = \frac{\pi}{3} \]

### Conclusion
Thus, the value of \( \theta \) is:
\[ \theta = \frac{\pi}{3} \]