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  4. The value of the expression cos 4 (π/8)+ cos 4 (3 π/8)+ cos 4 (5 π/8)+ cos 4 (7 π/8)=

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Q. The value of the expression $\cos ^{4} \frac{\pi}{8}+\cos ^{4} \frac{3 \pi}{8}+\cos ^{4} \frac{5 \pi}{8}+\cos ^{4} \frac{7 \pi}{8}=$

Solution:

$\cos ^{4} \frac{\pi}{8}+\cos ^{4} \frac{3 \pi}{8}+\cos ^{4} \frac{5 \pi}{8}+\cos ^{4} \frac{7 \pi}{8}$
$=2\left(\cos ^{4} \frac{\pi}{8}+\sin ^{4} \frac{\pi}{8}\right)=2\left[\left(\cos ^{2} \frac{\pi}{8}+\sin ^{2} \frac{\pi}{8}\right)^{2}-2 \cos ^{2} \frac{\pi}{8} \sin ^{2} \frac{\pi}{8}\right]$
$=2\left(1-\frac{1}{4}\right)=\frac{3}{2}$