If `theta` lies in the first quadrant and `cos^(2)theta - sin^(2) theta = (1)/(2)`, then the value of `tan^(2) 2 theta + sin^(2) 3 theta` is :

Options

A. `(7)/(2)`
B. 3
C. `(4)/(3)`
D. 4

Answer

To solve the problem step by step, we start with the equation given in the question:

1. **Given Equation**:
\[ \cos^2 \theta - \sin^2 \theta = \frac{1}{2} \]

2. **Using Trigonometric Identity**:
We know that:
\[ \cos^2 \theta - \sin^2 \theta = \cos 2\theta \]
Therefore, we can rewrite the equation as:
\[ \cos 2\theta = \frac{1}{2} \]

3. **Finding the Angle**:
The value of \( \cos 2\theta = \frac{1}{2} \) corresponds to:
\[ 2\theta = 60^\circ \quad \text{(since } \theta \text{ is in the first quadrant)} \]
Thus, we can solve for \( \theta \):
\[ \theta = 30^\circ \]

4. **Calculating \( \tan^2 2\theta \)**:
Now we need to find \( \tan^2 2\theta \):
\[ 2\theta = 60^\circ \implies \tan 2\theta = \tan 60^\circ = \sqrt{3} \]
Therefore:
\[ \tan^2 2\theta = (\sqrt{3})^2 = 3 \]

5. **Calculating \( \sin^2 3\theta \)**:
Next, we find \( \sin^2 3\theta \):
\[ 3\theta = 90^\circ \implies \sin 3\theta = \sin 90^\circ = 1 \]
Thus:
\[ \sin^2 3\theta = 1^2 = 1 \]

6. **Final Calculation**:
Now we can combine both results:
\[ \tan^2 2\theta + \sin^2 3\theta = 3 + 1 = 4 \]

7. **Conclusion**:
The final answer is:
\[ \boxed{4} \]