Question

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Find the value of below expression.

$\cos {}^{4} \frac{\pi}{8} + \cos {}^{4} \frac{3\pi}{8} + \cos {}^{4} \frac{5\pi}{8} + \cos {}^{4} \frac{7\pi}{8}$
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Answer 1

Given :-

$\displaystyle{\cos^4 \dfrac{\pi}{8}+\cos^4\dfrac{3\pi}{8}+\cos^4\dfrac{5\pi}{8}+\cos^4\dfrac{7\pi}{8}}$

Solution :-

5π/8 can be written as π - 3π/8 and 7π/8 can be written as π - π/8

$\sf\displaystyle \cos^4\dfrac{\pi}{8}+\cos^4\dfrac{3\pi}{8}+\cos^4\bigg({\pi - \dfrac{3\pi}{8}}\bigg) + \cos^4\bigg(\pi-\dfrac{\pi}{8}\bigg)$

$\displaystyle\cos^4\dfrac{\pi}{8}+\cos^4\dfrac{3\pi}{8}+\bigg(-\cos\dfrac{3\pi}{8}\bigg)^4 + \bigg(-\cos\dfrac{\pi}{8}\bigg)^4$

$\displaystyle \cos^4\dfrac{\pi}{8}+\cos^4\dfrac{3\pi}{8}\bigg - \bigg(\cos\dfrac{3\pi}{8}\bigg)^4 - \bigg(\cos\dfrac{\pi}{8}^4\bigg)$

$\displaystyle 2\bigg\{\cos^4\dfrac{\pi}{8}+\cos^4\dfrac{3\pi}{8}\bigg\}$

$\displaystyle 2\bigg\{\cos^4\dfrac{\pi}{8}+\sin^4\dfrac{\pi}{8}\bigg\}$

$\displaystyle 2\bigg\{\bigg(\cos^2\dfrac{\pi}{8}+\sin^2\dfrac{\pi}{8}\bigg)^2 - 2\cos^2\dfrac{\pi}{8}\times \sin^2\dfrac{\pi}{8}\bigg\}$

$\displaystyle 2\bigg\{\bigg(\cos^2\dfrac{\pi}{8}+\sin^2\dfrac{\pi}{8}\bigg)^2 - \bigg(2\sin\dfrac{\pi}{8}\cos\dfrac{\pi}{8}\bigg)^2\bigg\}$

$\sf 2\bigg\{\bigg(cos^2\dfrac{\pi}{8}+\sin^2\dfrac{\pi}{8}\bigg)^2 - \bigg(sin\dfrac{\pi}{4}\bigg)^2\bigg\}$

$\sf 2\bigg\{(1)^2-\dfrac{1}{2}\times\bigg(\dfrac{1}{\sqrt{2}}\bigg)^2\bigg\}$

$\sf 2\bigg\{1 - \dfrac{1}{2}\times\dfrac{1}{2}\bigg\}$

$\sf 2\bigg\{1 - \dfrac{1}{4}\bigg\}$

$\sf 2\bigg\{\dfrac{4-1}{4}\bigg\}$

$\sf 2\times\dfrac{3}{4}$

$\sf\dfrac{3}{2}$

Answer 2

Solution:

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

$= cos {}^{4} \frac{\pi}{8} + cos {}^{4} (\pi - \frac{3\pi}{8} ) + cos {}^{4} (\pi - \frac{\pi}{8} )$

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

$= 2[cos {}^{4} \frac{\pi}{8} + cos {}^{4} \frac{3\pi}{8} ]$

$= 2[cos {}^{4} \frac{\pi}{8} + cos {}^{4} ( \frac{\pi}{2} - \frac{\pi}{8} )]$

$= 2[cos {}^{4} \frac{\pi}{8} + cos {}^{4} \frac{\pi}{8} ]$

$= 2[(cos {}^{4} \frac{\pi}{8} + sin {}^{2} \frac{\pi}{8 } ) {}^{2} - 2cos {}^{2} \frac{\pi}{8 } \times \frac{\pi}{8} ]$

$= 2 - (2 - sin \: \frac{\pi}{8} \times cos \: \frac{\pi}{8} ) {}^{2}$

$= 2 - (sin \: \frac{\pi}{8} ) {}^{2}$

$= 2 - ( \frac{1}{ \sqrt{2} } ) {}^{2}$

$= 2 - \frac{1}{2} = \frac{3}{2}$

$\fbox\pink{Answer = 3/2}$