Question

The value of \( \sin^2 \left( \frac{3\pi}{8} \right) + \sin^2 \left( \frac{7\pi}{8} \right) \) is:

• A. \( \frac{1}{2} \)
• B. 1
• C. 3
• D. \( \frac{3}{4} \)
• E. \( \frac{1}{4} \)

Hint:

When calculating sums of squared trigonometric functions, consider using fundamental identities and angle relationships to simplify calculations.

Answer 1

The Correct Option is B

We start by recognizing the relationships between the angles involved: \[ \sin \left( \frac{7\pi}{8} \right) = \sin \left( \pi - \frac{\pi}{8} \right) = \sin \left( \frac{\pi}{8} \right) \] Therefore, we need to examine the relationship between \( \frac{3\pi}{8} \) and \( \frac{\pi}{8} \) using their complementary angles: \[ \sin \left( \frac{3\pi}{8} \right) = \cos \left( \frac{\pi}{8} \right) \] Given that \( \sin^2 \theta + \cos^2 \theta = 1 \) for any angle \( \theta \), substituting \( \frac{\pi}{8} \) into this identity gives: \[ \sin^2 \left( \frac{3\pi}{8} \right) + \sin^2 \left( \frac{7\pi}{8} \right) = \cos^2 \left( \frac{\pi}{8} \right) + \sin^2 \left( \frac{\pi}{8} \right) = 1 \]