cos^4 frac{pi}{8} + cos^4 frac{3pi}{8} + cos | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Given expression: $ S = \cos^4 \frac{\pi}{8} + \cos^4 \frac{3\pi}{8} + \cos^4 \frac{5\pi}{8} + \cos^4 \frac{7\pi}{8} $Since $ \cos(\pi - 	heta) =

Vedclass · Accepted Answer

$ \frac{3}{2} $

Vedclass · Answer

(C) Given expression: $ S = \cos^4 \frac{\pi}{8} + \cos^4 \frac{3\pi}{8} + \cos^4 \frac{5\pi}{8} + \cos^4 \frac{7\pi}{8} $Since $ \cos(\pi - 	heta) = -\cos 	heta $,we have $ \cos \frac{7\pi}{8} = -\cos \frac{\pi}{8} $ and $ \cos \frac{5\pi}{8} = -\cos \frac{3\pi}{8} $.Thus,$ \cos^4 \frac{7\pi}{8} = \cos^4 \frac{\pi}{8} $ and $ \cos^4 \frac{5\pi}{8} = \cos^4 \frac{3\pi}{8} $.Substituting these into the expression:$ S = 2 \left( \cos^4 \frac{\pi}{8} + \cos^4 \frac{3\pi}{8} ight) $Using $ a^2 + b^2 = (a+b)^2 - 2ab $:$ S = 2 \left[ \left( \cos^2 \frac{\pi}{8} + \cos^2 \frac{3\pi}{8} ight)^2 - 2 \cos^2 \frac{\pi}{8} \cos^2 \frac{3\pi}{8} ight] $Since $ \frac{3\pi}{8} = \frac{\pi}{2} - \frac{\pi}{8} $,$ \cos \frac{3\pi}{8} = \sin \frac{\pi}{8} $. Thus $ \cos^2 \frac{\pi}{8} + \cos^2 \frac{3\pi}{8} = \cos^2 \frac{\pi}{8} + \sin^2 \frac{\pi}{8} = 1 $.$ S = 2 \left[ 1^2 - 2 \cos^2 \frac{\pi}{8} \sin^2 \frac{\pi}{8} ight] = 2 \left[ 1 - \frac{1}{2} (2 \sin \frac{\pi}{8} \cos \frac{\pi}{8})^2 ight] $Using $ 2 \sin 	heta \cos 	heta = \sin 2	heta $:$ S = 2 \left[ 1 - \frac{1}{2} \sin^2 \frac{\pi}{4} ight] = 2 \left[ 1 - \frac{1}{2} \left( \frac{1}{\sqrt{2}} ight)^2 ight] = 2 \left[ 1 - \frac{1}{4} ight] = 2 \left( \frac{3}{4} ight) = \frac{3}{2} $.

$ \cos^4 \frac{\pi}{8} + \cos^4 \frac{3\pi}{8} + \cos^4 \frac{5\pi}{8} + \cos^4 \frac{7\pi}{8} = $

Similar Questions

Which one of the following gases can deplete the ozone layer in the upper atmosphere?

Which country produces the maximum amount of greenhouse gases?

How many different types of gametes will be produced by a plant with the genotype $AABbCc$?

Gametophyte is the dominant phase in the life cycle of

If $f(x) = \frac{x^2-10x+25}{x^2-7x+10}$ and $f$ is continuous at $x = 5$,then $f(5)$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module