Examples

Chapter 3 Class 11 Trigonometric Functions
Serial order wise

### Transcript

Example 22 Prove that cos2 ๐ฅ+cos2 (๐ฅ+๐/3) + cos2 (๐ฅโ๐/3) = 3/2 Lets first calculate all 3 terms separately We know that cos 2x = 2 cos2 x โ 1 cos 2x + 1 = 2cos2 x ๐๐๐ โกใ2๐ฅ + 1ใ/2 = cos2 x cos2 x = ๐๐จ๐ฌโกใ๐๐ + ๐ใ/๐ Replacing x with ("x + " ๐/๐) cos2 ("x" +๐/๐) = cosโกใ2(๐ฅ + ๐/3)+1ใ/2 = ๐๐๐โกใ(๐๐ + ๐๐/๐) + ๐ใ/๐ Similarly, Replacing x with ("x โ" ๐/๐) in cos2 x = ๐๐จ๐ฌโกใ๐๐ + ๐ใ/๐ cos2 ("x" โ๐/3) = cosโกใ2(๐ฅ โ ๐/3)+ 1ใ/2 = cosโกใ(2๐ฅ โ 2๐/3)+ 1ใ/2 Solving L.H.S cos2 x + cos2 (๐ฅ+ ๐/3) + cos2 (๐ฅโ๐/3) = (๐ + ๐๐จ๐ฌโก๐๐)/๐ + (๐ + ๐๐๐โก(๐๐ + ๐๐/๐))/๐ + (๐ + ๐๐๐โก(๐๐ โ ๐๐/๐))/๐ = 1/2 [1+cosโกใ2๐ฅ+1+๐๐๐ (2๐ฅ+2๐/3)+1+๐๐๐ (2๐ฅโ2๐/3)ใ ] = 1/2 [3+cosโกใ2๐ฅ+๐๐๐(๐๐+๐๐/๐)+๐๐๐(๐๐โ๐๐/๐)ใ ] = 1/2 [3+cosโกใ2๐ฅ+2๐๐๐((๐๐ + ๐๐/๐ + ๐๐ โ ๐๐/๐)/๐).๐๐๐((๐๐ + ๐๐/๐ โ(๐๐ โ ๐๐/๐))/๐)ใ ] = 1/2 [3+cosโกใ2๐ฅ+2๐๐๐ ((4๐ฅ + 0)/2).๐๐๐ ((0 + 4๐/3)/2)ใ ] = 1/2 [3+cosโกใ2๐ฅ+2๐๐๐ (4๐ฅ/2).๐๐๐ ((4๐/3)/2)ใ ] = 1/2 [3+cosโกใ2๐ฅ+๐ ๐๐จ๐ฌโก๐๐ ๐๐จ๐ฌโกใ๐๐/๐ใ ใ ] = 1/2 [3+cosโกใ2๐ฅ+2 cosโก2๐ฅ cosโก(๐โ๐/3) ใ ] = 1/2 [3+cosโกใ2๐ฅ+2 cosโก2๐ฅ ใ (ใโ๐๐๐ใโก(๐/๐) ) ] = 1/2 [3+cosโกใ2๐ฅ+2 cosโก2๐ฅ ใ (โ1/2) ] = 1/2 [3+cosโกใ2๐ฅโ2 ร1/2รcosโก2๐ฅ ใ ] = ๐/๐ [๐+๐๐๐โกใ๐๐โ๐๐๐โก๐๐ ใ ] = 1/2 [3+0] = 3/2 = R.H.S. Hence, L.H.S. = R.H.S. Hence Proved

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#### Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.