Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# Prove That: Cos 2 π 8 + Cos 2 3 π 8 + Cos 2 5 π 8 + Cos 2 7 π 8 = 2 - Mathematics

Numerical

Prove that: $\cos^2 \frac{\pi}{8} + \cos^2 \frac{3\pi}{8} + \cos^2 \frac{5\pi}{8} + \cos^2 \frac{7\pi}{8} = 2$

#### Solution

$LHS = \cos^2 \frac{\pi}{8} + \cos^2 \frac{3\pi}{8} + \cos^2 \frac{5\pi}{8} + \cos^2 \frac{7\pi}{8}$

$= \cos^2 \frac{\pi}{8} + \cos^2 \frac{3\pi}{8} + \cos^2 \left( \pi - \frac{3\pi}{8} \right) + \cos^2 \left( \pi - \frac{\pi}{8} \right)$

$= \cos^2 \frac{\pi}{8} + \cos^2 \frac{3\pi}{8} + \left\{ - \cos\left( \frac{3\pi}{8} \right) \right\}^2 + \left\{ - \cos\left( \frac{\pi}{8} \right) \right\}^2$

$= \cos^2 \frac{\pi}{8} + \cos^2 \frac{3\pi}{8} + \cos^2 \frac{3\pi}{8} + \cos^2 \frac{\pi}{8}$

$= 2\left( \cos^2 \frac{\pi}{8} + \cos^2 \frac{3\pi}{8} \right)$

$= 2\left\{ \cos^2 \frac{\pi}{8} + \cos^2 \left( \frac{\pi}{2} - \frac{\pi}{8} \right) \right\}$

$= 2\left( \cos^2 \frac{\pi}{8} + \sin^2 \frac{\pi}{8} \right)$

$= 2 = RHS$

$\text{ Hence proved } .$

Concept: Values of Trigonometric Functions at Multiples and Submultiples of an Angle
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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 9 Values of Trigonometric function at multiples and submultiples of an angle
Exercise 9.1 | Q 9 | Page 28