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Question
Find the value of the expression `cos^4 pi/8 + cos^4 (3pi)/8 + cos^4 (5pi)/8 + cos^4 (7pi)/8`
[Hint: Simplify the expression to `2(cos^4 pi/8 + cos^4 (3pi)/8) = 2[(cos^2 pi/8 + cos^2 (3pi)/8)^2 - 2cos^2 pi/8 cos^2 (3pi)/8]`
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Solution
`cos^4 pi/8 + cos^4 (3pi)/8 + cos^4 (5pi)/8 + cos^4 (7pi)/8`
= `cos^4 pi/8 + cos^4 (3pi)/8 + cos^4 (pi - (3pi)/8) + cos^4 (pi - pi/8)`
= `cos^4 pi/8 + cos^4 (3pi)/8 + cos^4 (3pi)/8 + cos^4 pi/8`
= `2cos^4 pi/8 + 2cos^4 (3pi)/8`
= `2[cos^4 pi/8 + cos^4 (3pi)/8]`
= `2[cos^4 pi/8 + cos^4 (pi/2 - pi/8)]`
= `2[cos^4 pi/8 + sin^4 pi/8]`
= `2[cos^4 pi/8 + sin^4 pi/8 + 2sin^2 pi/8 . cos^2 pi/8 - 2sin^2 pi/8 . cos^2 pi/8]`
= `2[(cos^2 pi/8 + sin^2 pi/8)^2 - 2sin^2 pi/8 . cos^2 pi/8]`
= `2[1 - 2sin^2 pi/8 cos^2 pi/8]`
= `2 - 4sin^2 pi/8 . cos^2 pi/8`
= `2 - (2sin pi/8 . cos pi/8)^2`
= `2 - (sin pi/4)^2`
= `2 - (1/sqrt(2))^2`
= `2 - 1/2`
= `3/2`
Hence, the required value of the expression = `3/2`
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