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