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  • 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}-

Find the value of the expression \cos\textsuperscript{4}( \pi /8) + \cos\textsuperscript{4}(3 \pi /8) + \cos\textsuperscript{4}(5 \pi /8) + \cos\textsuperscript{4}(7 \pi /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]

Answers (1)

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


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


 

 

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