What is cos^2(pi/7)+cos^2(2pi/7)+cos^2(4pi/7) is equal to?

leshumict6

leshumict6

Answered question

2023-02-15

What is cos 2 ( π 7 ) + cos 2 ( 2 π 7 ) + cos 2 ( 4 π 7 ) is equal to?

Answer & Explanation

Genesis Hancock

Genesis Hancock

Beginner2023-02-16Added 10 answers

We have: LHS = cos 2 ( π 7 ) + cos 2 ( 2 π 7 ) + cos 2 ( 4 π 7 )
using formula cos 2 θ = 1 2 ( 1 + cos 2 θ )
= 1 2 ( 1 + cos ( 2 π 7 ) ) + 1 2 ( 1 + cos ( 4 π 7 ) ) + 1 2 ( 1 + cos ( 8 π 7 ) )
= 3 2 + 1 2 ( cos ( 2 π 7 ) + cos ( 4 π 7 ) + cos ( 8 π 7 ) )
= 3 2 + 1 4 sin ( π 7 ) ( 2 sin ( π 7 ) cos ( 2 π 7 ) + 2 sin ( π 7 ) cos ( 4 π 7 ) + 2 sin ( π 7 ) cos ( 8 π 7 ) )
= 3 2 + 1 4 sin ( π 7 ) ( sin ( 3 π 7 ) - sin ( π 7 ) + sin ( 5 π 7 ) - sin ( 3 π 7 ) + sin ( 9 π 7 ) - sin ( 7 π 7 ) )
= 3 2 + 1 4 sin ( π 7 ) ( - sin ( π 7 ) + sin ( π - 2 π 7 ) + sin ( π + 2 π 7 ) - 0 )
= 3 2 + 1 4 sin ( π 7 ) ( - sin ( π 7 ) + sin ( 2 π 7 ) - sin ( 2 π 7 ) )
= 3 2 + 1 4 sin ( π 7 ) ( - sin ( π 7 ) )
= 3 2 - 1 4 = 6 - 1 4 = 5 4

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