Trigonometric Ratios and Identities
Complementary Angles
JEE Main 2020
Grade 11
Question:
$\cos^3 \frac{\pi}{8} \cos \frac{3\pi}{8} + \sin^3 \frac{\pi}{8} \sin \frac{3\pi}{8}$ equals:
(1) $\frac{1}{2\sqrt{2}}$
(2) $\frac{\sqrt{2}}{8}$
(3) $\frac{1}{4}$
(4) $\frac{\sqrt{2}}{4}$
Step-by-Step Solution
Key Concept: Note $3\pi/8 = \pi/2 - \pi/8$, so $\cos(3\pi/8) = \sin(\pi/8)$ and $\sin(3\pi/8) = \cos(\pi/8)$. Factor out $\sin(\pi/8) \cos(\pi/8)$ and use the double-angle identity.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)