The value of cos4⁡π8+cos4⁡3π8+ cos4⁡5π8+cos4⁡7π8 is equal to

We can write the given expression ascos4⁡π8+cos4⁡3π8+cos4⁡π−3π8+cos4⁡π−π8=2cos4⁡π8+cos4⁡3π8=2cos4⁡π8+sin4⁡π2−3π8=2cos4⁡π8+sin4⁡π8=2cos2⁡π8+sin2⁡π82−2cos2⁡π8sin2⁡π8=21−12sin2⁡π4=21−12×12=32

The value of cos4⁡π8+cos4⁡3π8+ cos4⁡5π8+cos4⁡7π8 is equal to

We can write the given expression ascos4⁡π8+cos4⁡3π8+cos4⁡π−3π8+cos4⁡π−π8=2cos4⁡π8+cos4⁡3π8=2cos4⁡π8+sin4⁡π2−3π8=2cos4⁡π8+sin4⁡π8=2cos2⁡π8+sin2⁡π82−2cos2⁡π8sin2⁡π8=21−12sin2⁡π4=21−12×12=32

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The value of $\cos^{4} (\frac{π}{8}) + \cos^{4} (\frac{3 π}{8}) +$ $\cos^{4} (\frac{5 π}{8}) + \cos^{4} (\frac{7 π}{8})$ is equal to

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$1 / 2$

$3 / 4$

$3 / 2$

answer is C.

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Detailed Solution

We can write the given expression as

$\begin{array}{l} \cos^{4} (\frac{π}{8}) + \cos^{4} (\frac{3 π}{8}) + \cos^{4} (π - \frac{3 π}{8}) + \cos^{4} (π - \frac{π}{8}) \\ = 2 [\cos^{4} (\frac{π}{8}) + \cos^{4} (\frac{3 π}{8})] \\ = 2 [\cos^{4} (\frac{π}{8}) + \sin^{4} (\frac{π}{2} - \frac{3 π}{8})] \\ = 2 [\cos^{4} (\frac{π}{8}) + \sin^{4} \frac{π}{8}] \\ = 2 [{(\cos^{2} (\frac{π}{8}) + \sin^{2} (\frac{π}{8}))}^{2} - 2 \cos^{2} (\frac{π}{8}) \sin^{2} (\frac{π}{8})] \end{array}$