If `0lethetale(pi)/(2)andsec^(2)theta+tan^(2)theta=7` , then `theta` is

Options

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

Answer

To solve the problem step by step, we will follow the instructions given in the video transcript.

### Step 1: Understand the given equation
We have the equation:
\[ \sec^2 \theta + \tan^2 \theta = 7 \]

### Step 2: Use the Pythagorean identity
We know the Pythagorean identity:
\[ \sec^2 \theta - \tan^2 \theta = 1 \]

### Step 3: Set up the equations
Now we have two equations:
1. \( \sec^2 \theta + \tan^2 \theta = 7 \) (Equation 1)
2. \( \sec^2 \theta - \tan^2 \theta = 1 \) (Equation 2)

### Step 4: 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 5: Solve for \(\sec^2 \theta\)
Dividing both sides by 2:
\[ \sec^2 \theta = 4 \]

### Step 6: Find \(\sec \theta\)
Taking the square root of both sides:
\[ \sec \theta = \sqrt{4} = 2 \]
Since \(\theta\) is in the first quadrant, we only consider the positive value.

### Step 7: Find \(\cos \theta\)
Recall that \(\sec \theta = \frac{1}{\cos \theta}\), thus:
\[ \cos \theta = \frac{1}{\sec \theta} = \frac{1}{2} \]

### Step 8: Find \(\theta\)
Now, we need to find \(\theta\) such that:
\[ \cos \theta = \frac{1}{2} \]
This corresponds to:
\[ \theta = 60^\circ \]

### Step 9: Convert to radians
To convert degrees to radians:
\[ \theta = 60^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{3} \text{ radians} \]

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

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