If `sec^(2)theta+tan^(2)theta=7` , then the value of `theta`

Options

A. `60^(@)`
B. `30^(@)`
C. `0^(@)`
D. `90^(@)`

Answer

To solve the equation \( \sec^2 \theta + \tan^2 \theta = 7 \), we can use the trigonometric identity that relates secant and tangent:

1. **Recall the identity**:
\[ \sec^2 \theta = 1 + \tan^2 \theta \]

2. **Substitute the identity into the equation**:
\[ \sec^2 \theta + \tan^2 \theta = (1 + \tan^2 \theta) + \tan^2 \theta = 1 + 2\tan^2 \theta \]

3. **Set the equation equal to 7**:
\[ 1 + 2\tan^2 \theta = 7 \]

4. **Isolate \( \tan^2 \theta \)**:
\[ 2\tan^2 \theta = 7 - 1 \]
\[ 2\tan^2 \theta = 6 \]
\[ \tan^2 \theta = \frac{6}{2} = 3 \]

5. **Take the square root to find \( \tan \theta \)**:
\[ \tan \theta = \sqrt{3} \quad \text{or} \quad \tan \theta = -\sqrt{3} \]

6. **Find the angles corresponding to \( \tan \theta = \sqrt{3} \)**:
\[ \theta = 60^\circ + n \cdot 180^\circ \quad \text{for } n \in \mathbb{Z} \]

7. **Find the angles corresponding to \( \tan \theta = -\sqrt{3} \)**:
\[ \theta = 120^\circ + n \cdot 180^\circ \quad \text{for } n \in \mathbb{Z} \]

Thus, the values of \( \theta \) can be \( 60^\circ \) or \( 120^\circ \) plus any integer multiple of \( 180^\circ \).