Question

The value of \( \sin^2 \frac{\pi}{8} + \sin^2 \frac{3\pi}{8} + \sin^2 \frac{5\pi}{8} + \sin^2 \frac{7\pi}{8} \) is equal to:

• A. $\frac{1}{8}$
• B. $\frac{1}{4}$
• C. $\frac{1}{2}$
• D. $1$
• E. $2$

Hint:

In sums of $\sin^2$ or $\cos^2$ with symmetric arguments, try to pair the first and last terms. They almost always simplify to 1 using complementary or supplementary relationships.

Answer 1

Answer: (E)

Concept: Use trigonometric identities to simplify terms, specifically the supplementary angle identity $\sin(\pi - \theta) = \sin \theta$ and the complementary angle identity $\sin(\frac{\pi}{2} - \theta) = \cos \theta$.
• $\sin^2 \theta + \cos^2 \theta = 1$

Step 1: Reduce the angles using supplementary identities.
Observe that $\frac{7\pi}{8} = \pi - \frac{\pi}{8}$ and $\frac{5\pi}{8} = \pi - \frac{3\pi}{8}$. \[ \sin^2 \frac{7\pi}{8} = \sin^2 (\pi - \frac{\pi}{8}) = \sin^2 \frac{\pi}{8} \] \[ \sin^2 \frac{5\pi}{8} = \sin^2 (\pi - \frac{3\pi}{8}) = \sin^2 \frac{3\pi}{8} \] The expression becomes: \[ 2 \left( \sin^2 \frac{\pi}{8} + \sin^2 \frac{3\pi}{8} \right) \]

Step 2: Use the complementary angle identity.
Observe that $\frac{3\pi}{8} = \frac{\pi}{2} - \frac{\pi}{8}$. \[ \sin \frac{3\pi}{8} = \sin \left( \frac{\pi}{2} - \frac{\pi}{8} \right) = \cos \frac{\pi}{8} \] So, $\sin^2 \frac{3\pi}{8} = \cos^2 \frac{\pi}{8}$.

Step 3: Evaluate the final sum.
\[ 2 \left( \sin^2 \frac{\pi}{8} + \cos^2 \frac{\pi}{8} \right) = 2(1) = 2 \]