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1 solution
Let S = cos 2 A + cos 2 ( A + 1 2 0 ) + cos 2 ( A − 2 ) ].
Then 2 S = 2 cos 2 A + 2 cos 2 ( A + 1 2 0 ) + 2 cos 2 ( A − 2 ) ⇒ 2 S = 2 cos 2 A + cos ( 2 A + 2 4 0 ) + 1 + cos ( 2 A − 2 4 0 ) + 1 ⇒ 2 S = 2 cos 2 a + 1 − cos 2 A + 2 ⇒ 2 S = 3 ⇒ s = 3 / 2
Such transformations are important in many identities involving angles of a triangle.