Find the value of the expression

Question:

Find the value of the expression cos4(π/8) + cos4(3π/8) + cos4(5π/8) + cos4(7π/8).
[Hint: Simplify the expression to

$2\left(\cos ^{4} \frac{\pi}{8}+\cos ^{4} \frac{3 \pi}{8}\right)=2\left[\left(\cos ^{2} \frac{\pi}{8}+\cos ^{2} \frac{3 \pi}{8}\right)^{2}-2 \cos ^{2} \frac{\pi}{8} \cos ^{2} \frac{3 \pi}{8}\right]$

Solution:

According to the question,

Let y = cos4(π/8) + cos4(3π/8) + cos4(5π/8) + cos4(7π/8).

⇒ y = cos4(π/8) + cos4(3π/8) + cos4(π – 3π/8) + cos4(π – π/8).

Since we know that, cos (π – x) = – cos x, we get,

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

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

Since, we know that,

$\cos (\pi-x)=-\cos x$

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

$=2\left(\cos ^{4} \frac{\pi}{8}+\cos ^{4} \frac{3 \pi}{8}\right)$

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

$=2\left(\cos ^{4} \frac{\pi}{8}+\sin ^{4} \frac{\pi}{8}\right)$

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

$=2\left[1-2 \cos ^{2} \frac{\pi}{8} \cdot \sin ^{2} \frac{\pi}{8}\right]$

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

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

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

$=2-1 / 2$

$=3 / 2$

= 2 – (1/√2)2

= 2 – ½

= 3/2