1.

Evaluate: (\(1+cos\frac{\pi}{8}\)) (\(1+cos\frac{3\pi}{8}\)) (\(1+cos\frac{5\pi}{8}\)) (\(1+cos\frac{7\pi}{8}\))

Answer»

We have, 

(\(1+cos \frac{\pi}{8}\) ) (\(1+cos\frac{3\pi}{8}\) ) (\(1+cos\frac{5\pi}{8}\) ) (\(1+cos\frac{7\pi}{8}\)

= (\(1+cos\frac{\pi}{8}\)) (\(1+sin(\frac{\pi}{2}-\frac{3\pi}{8})\)) (\(1+sin(\frac{\pi}{2}-\frac{5\pi}{8})\)) (\(1+cos(\pi -\frac{\pi}{8})\)

= (\(1+cos\frac{\pi}{8}\)) (\(1+sin\frac{\pi}{8}\)) . (\(1-sin\frac{\pi}{8}\)) (\(1-cos\frac{\pi}{8}\)

= (\(1-cos^2\frac{\pi}{8}\) ) (\(1-sin^2\frac{\pi}{8}\)

= \(sin^2\frac{\pi}{8}cos^2\frac{\pi}{8}\) 

= \(\frac{1}{4}(2sin\frac{\pi}{8}cos\frac{\pi}{8})^2\) 

=\(\frac{1}{4}sin^2(\frac{2\pi}{8})\)

= \(\frac{1}{4}sin^2(\frac{2\pi}{8})\)

=\(\frac{1}{4}(\frac{1}{\sqrt{2}})^2\) 

= \(\frac{1}{4}\times \frac{1}{2}\)\(\frac{1}{8}\)



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