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Question Number 168297 by MdNafiz last updated on 07/Apr/22

prove that \dots \dots . .

\cos \frac{2 Π}{7} + \cos \frac{4 Π}{7} + \cos \frac{8 Π}{7} = - \frac{1}{2}

Answered by som(math1967) last updated on 07/Apr/22

l e t \frac{π}{7} = θ

∴ c o s 2 θ + c o s 4 θ + c o s 8 θ

= \frac{2 s i n 2 θ c o s 2 θ + 2 s i n 2 θ c o s 4 θ + 2 s i n 2 θ c o s 8 θ}{2 s i n 2 θ}

= \frac{s i n 4 θ + s i n 6 θ - s i n 2 θ + s i n 10 θ - s i n 6 θ}{2 s i n 2 θ}

= \frac{s i n 4 θ + s i n 10 θ - s i n 2 θ}{2 s i n 2 θ}

= \frac{2 s i n 7 θ c o s 3 θ - s i n 2 θ}{2 s i n 2 θ}

= \frac{0 - s i n 2 θ}{2 s i n 2 θ} = - \frac{1}{2} ★

★ s i n 7 θ = s i n π = 0

Commented by peter frank last updated on 08/Apr/22

good