Last updated at Feb. 13, 2020 by Teachoo

Transcript

Misc 1 Prove that: 2cos ๐/13 cos 9๐/13 + cos 3๐/13 + cos 5๐/13 = 0 Taking L.H.S 2cos ๐/13 cos 9๐/13 + cos 3๐/13 + cos 5๐/13 = ("cos " 10๐/13 " + cos " 8๐/13) + cos 3๐/13 + cos 5๐/13 We know that 2 cos x cos y = cos (x + y) + cos (x โ y) Putting x = 9๐/13 and y = ๐/13 2cos 9๐/13 cos ๐/13 = cos (9๐/13 " + " ๐/13) + cos(9๐/13 " + " ๐/13) = cos (10๐/13) + cos ((8 ๐)/13) = ("cos " 10๐/13 " + cos " 3๐/13) + ("cos " 8๐/13 " + cos " 5๐/13) = ("2 cos " ((10๐/13 + 3๐/13)/2)" . cos " ((10๐/13 โ 3๐/13)/2)) + ("2cos " ((8๐/13 + 5๐/13)/2)" . cos " ((8๐/13 โ 5๐/13)/2)) = ("2 cos " ((13๐/13)/2)" . cos " ((7๐/13)/2)) + ("2 cos " (13๐/13)/2 " . cos " (3๐/13)/2) = ("2 cos " ๐/2 " . cos " 7๐/26) + ("2 cos " ๐/2 " . cos " 3๐/26) = 2 cos ๐/2 ("cos " 7๐/26 " + cos " 3๐/26) = 2 ร 0 ("cos " 7๐/26 " + cos " 3๐/26) = 0 = R.H.S. Hence L.H.S. = R.H.S. Hence proved (cos ๐/2 = 0)

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.