A) \[\frac{7}{2}\]
B) \[\cos \frac{\pi }{8}\]
C) \[\frac{1}{8}\]
D) \[\frac{1+\sqrt{2}}{2\sqrt{2}}\]
Correct Answer: C
Solution :
\[\left( 1+\cos \frac{\pi }{8} \right)\left( 1+\cos \frac{3\pi }{8} \right)\left( 1+\cos \frac{5\pi }{8} \right)\] \[\left( 1+\cos \frac{7\pi }{8} \right)\] \[=\left( 1+\cos \frac{\pi }{8} \right)\left( 1+\cos \frac{3\pi }{8} \right)\left( 1-\cos \frac{3\pi }{8} \right)\] \[\left( 1-\cos \frac{\pi }{8} \right)\] \[=\left( 1-{{\cos }^{2}}\frac{\pi }{8} \right)\left( 1-{{\cos }^{2}}\frac{3\pi }{8} \right)\] \[=\left[ 1-\left( \frac{1+\cos \frac{\pi }{4}}{2} \right) \right]\left[ 1-\frac{\left( 1+\cos \frac{3\pi }{4} \right)}{2} \right]\] \[=\frac{1}{4}\left( 1-\cos \frac{\pi }{4} \right)\left( 1-\cos \frac{3\pi }{4} \right)\] \[=\frac{1}{4}\left( 1-\cos \frac{\pi }{4} \right)\left( 1-\cos \frac{3\pi }{4} \right)\] \[=\frac{1}{4}\left( 1-\frac{1}{\sqrt{2}} \right)\left( 1+\frac{1}{\sqrt{2}} \right)\] \[=\frac{1}{4}\left( 1-\frac{1}{2} \right)=\frac{1}{8}\]
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