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Question
Find the value of `(1 + cos pi/8)(1 + cos (3pi)/8)(1 + cos (5pi)/8)(1 + cos (7pi)/8)`
Solution
Write `(1 + cos pi/8)(1 + cos (3pi)/8)(1 + cos (5pi)/8)(1 + cos (7pi)/8)`
= `(1 + cos pi/8)(1 + cos (3pi)/8)(1 + cos (pi - (3pi)/8))(1 + cos(pi - pi/8))`
= `(1 - cos^2 pi/8)(1 - cos^2 (3pi)/8)`
= `sin^2 pi/8 sin^2 (3pi)/8`
= `1/4 (1 - cos pi/4)(1 - cos (3pi)/4)`
= `1/4 (1 - cos pi/4)(1 + cos pi/4)`
= `1/4 (1 - cos^2 pi/4)`
= `1/4(1 - 1/2)`
= `1/8`
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