Slide64.JPG

Slide65.JPG
Slide66.JPG Slide67.JPG Slide68.JPG

This video is only available for Teachoo black users

Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


Transcript

Example 22 Prove that cos2 ๐‘ฅ+cos2 (๐‘ฅ+๐œ‹/3) + cos2 (๐‘ฅโˆ’๐œ‹/3) = 3/2 Lets first calculate all 3 terms separately We know that cos 2x = 2 cos2 x โˆ’ 1 cos 2x + 1 = 2cos2 x ๐‘๐‘œ๐‘ โกใ€–2๐‘ฅ + 1ใ€—/2 = cos2 x cos2 x = ๐œ๐จ๐ฌโกใ€–๐Ÿ๐’™ + ๐Ÿใ€—/๐Ÿ Replacing x with ("x + " ๐…/๐Ÿ‘) cos2 ("x" +๐…/๐Ÿ‘) = cosโกใ€–2(๐‘ฅ + ๐œ‹/3)+1ใ€—/2 = ๐’„๐’๐’”โกใ€–(๐Ÿ๐’™ + ๐Ÿ๐…/๐Ÿ‘) + ๐Ÿใ€—/๐Ÿ Similarly, Replacing x with ("x โˆ’" ๐…/๐Ÿ‘) in cos2 x = ๐œ๐จ๐ฌโกใ€–๐Ÿ๐’™ + ๐Ÿใ€—/๐Ÿ cos2 ("x" โˆ’๐œ‹/3) = cosโกใ€–2(๐‘ฅ โˆ’ ๐œ‹/3)+ 1ใ€—/2 = cosโกใ€–(2๐‘ฅ โˆ’ 2๐œ‹/3)+ 1ใ€—/2 Solving L.H.S cos2 x + cos2 (๐‘ฅ+ ๐œ‹/3) + cos2 (๐‘ฅโˆ’๐œ‹/3) = (๐Ÿ + ๐œ๐จ๐ฌโก๐Ÿ๐’™)/๐Ÿ + (๐Ÿ + ๐’„๐’๐’”โก(๐Ÿ๐’™ + ๐Ÿ๐…/๐Ÿ‘))/๐Ÿ + (๐Ÿ + ๐’„๐’๐’”โก(๐Ÿ๐’™ โˆ’ ๐Ÿ๐…/๐Ÿ‘))/๐Ÿ = 1/2 [1+cosโกใ€–2๐‘ฅ+1+๐‘๐‘œ๐‘ (2๐‘ฅ+2๐œ‹/3)+1+๐‘๐‘œ๐‘ (2๐‘ฅโˆ’2๐œ‹/3)ใ€— ] = 1/2 [3+cosโกใ€–2๐‘ฅ+๐’„๐’๐’”(๐Ÿ๐’™+๐Ÿ๐…/๐Ÿ‘)+๐’„๐’๐’”(๐Ÿ๐’™โˆ’๐Ÿ๐…/๐Ÿ‘)ใ€— ] = 1/2 [3+cosโกใ€–2๐‘ฅ+2๐’„๐’๐’”((๐Ÿ๐’™ + ๐Ÿ๐…/๐Ÿ‘ + ๐Ÿ๐’™ โˆ’ ๐Ÿ๐…/๐Ÿ‘)/๐Ÿ).๐’„๐’๐’”((๐Ÿ๐’™ + ๐Ÿ๐…/๐Ÿ‘ โˆ’(๐Ÿ๐’™ โˆ’ ๐Ÿ๐…/๐Ÿ‘))/๐Ÿ)ใ€— ] = 1/2 [3+cosโกใ€–2๐‘ฅ+2๐‘๐‘œ๐‘ ((4๐‘ฅ + 0)/2).๐‘๐‘œ๐‘ ((0 + 4๐œ‹/3)/2)ใ€— ] = 1/2 [3+cosโกใ€–2๐‘ฅ+2๐‘๐‘œ๐‘ (4๐‘ฅ/2).๐‘๐‘œ๐‘ ((4๐œ‹/3)/2)ใ€— ] = 1/2 [3+cosโกใ€–2๐‘ฅ+๐Ÿ ๐œ๐จ๐ฌโก๐Ÿ๐’™ ๐œ๐จ๐ฌโกใ€–๐Ÿ๐…/๐Ÿ‘ใ€— ใ€— ] = 1/2 [3+cosโกใ€–2๐‘ฅ+2 cosโก2๐‘ฅ cosโก(๐œ‹โˆ’๐œ‹/3) ใ€— ] = 1/2 [3+cosโกใ€–2๐‘ฅ+2 cosโก2๐‘ฅ ใ€— (ใ€–โˆ’๐’„๐’๐’”ใ€—โก(๐…/๐Ÿ‘) ) ] = 1/2 [3+cosโกใ€–2๐‘ฅ+2 cosโก2๐‘ฅ ใ€— (โˆ’1/2) ] = 1/2 [3+cosโกใ€–2๐‘ฅโˆ’2 ร—1/2ร—cosโก2๐‘ฅ ใ€— ] = ๐Ÿ/๐Ÿ [๐Ÿ‘+๐’„๐’๐’”โกใ€–๐Ÿ๐’™โˆ’๐’„๐’๐’”โก๐Ÿ๐’™ ใ€— ] = 1/2 [3+0] = 3/2 = R.H.S. Hence, L.H.S. = R.H.S. Hence Proved

Ask a doubt
Davneet Singh's photo - Co-founder, Teachoo

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.