Learn All Concepts of Chapter 2 Class 11 Relations and Function - FREE. Check - Trigonometry Class 11 - All Concepts

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  1. Chapter 3 Class 11 Trigonometric Functions
  2. Serial order wise

Transcript

Example 26 Prove that cos 2x cos ๐‘ฅ/2 โ€“ cos 3x cos 9๐‘ฅ/2 = sin 5x sin 5๐‘ฅ/2 Solving L.H.S Solving cos 2x cos x/2 and cos 3x cos 9๐‘ฅ/2 separately We know that 2 cos x cos y = cos (x + y) + cos (x โ€“ y) cos x cos y = 1/2 ("cos (x + y) + cos (x โ€“ y)" ) cos 2x cos ๐’™/๐Ÿ Replacing x with 2x and y with ๐‘ฅ/2 = 1/2 ("cos " ("2x + " x/2)" + cos" ("2x" โˆ’x/2)) = 1/2 ("cos " ((4x + x )/2)" + cos " ((4x โˆ’ x)/2)) = 1/2 ("cos " (5x/2)" + cos " (3x/2)) cos 3x cos ๐Ÿ—๐’™/๐Ÿ Replacing x with 3x and y with 9๐‘ฅ/2 = 1/2 ("cos" ("3x + " 9๐‘ฅ/2)" + cos " ("3x โ€“ " 9๐‘ฅ/2)) = 1/2 ("cos " ((6๐‘ฅ + 9๐‘ฅ )/2)" + cos " ((6๐‘ฅ โˆ’ 9๐‘ฅ)/2)) = 1/2 ("cos " (15๐‘ฅ/2)" + cos " ((โˆ’3๐‘ฅ)/2)) Now, cos 2x cos ๐‘ฅ/2 โ€“ cos 3x cos 9๐‘ฅ/2 Putting values = 1/2 ("cos " (5x/2)" + cos " (3x/2)) โ€“ 1/2 ("cos " (15๐‘ฅ/2)" + cos " ((โˆ’3๐‘ฅ)/2)) = 1/2 ("cos " (5x/2)" + cos " (3x/2)) โ€“ 1/2 ("cos " (15๐‘ฅ/2)" + cos " (3๐‘ฅ/2)) = 1/2 ("cos " (5x/2)" + cos " (3x/2)"โ€“ cos " (15๐‘ฅ/2)"โ€“ cos " (3๐‘ฅ/2)) = 1/2 ("cos " (5x/2)"โ€“ cos " (15๐‘ฅ/2)"+ cos " (3x/2)"โ€“ cos " (3๐‘ฅ/2)) = 1/2 ("cos " (5x/2)"โ€“ cos " (15๐‘ฅ/2)"+ " 0) = 1/2 ("cos " (5x/2)"โ€“ cos " (15๐‘ฅ/2)) = 1/2 ("โ€“ 2 sin " ((5๐‘ฅ/2 +15/2 ๐‘ฅ)/2)". sin " ((5๐‘ฅ/2 โˆ’ 15/2 ๐‘ฅ)/2)) = โˆ’"sin " ((5๐‘ฅ + 15๐‘ฅ)/(2 ร— 2))" . sin " ((5๐‘ฅ โˆ’ 15๐‘ฅ)/(2 ร— 2)) = "โ€“ sin " (20๐‘ฅ/4)" . sin " (( โˆ’10๐‘ฅ)/4) Using cos x โ€“ cos y = โ€“2 sin (๐‘ฅ + ๐‘ฆ)/2 sin (๐‘ฅ โˆ’ ๐‘ฆ)/2 Putting x = 5๐‘ฅ/2 , y = 15๐‘ฅ/2 = โˆ’ "sin " (5๐‘ฅ)" . sin " (( โˆ’5๐‘ฅ)/2) = โˆ’ "sin " (5๐‘ฅ) ร— โˆ’" sin " (( 5๐‘ฅ)/2) = sin (5๐‘ฅ) sin (( 5๐‘ฅ)/2) = R.H.S Hence L.H.S. = R.H.S. Hence proved (As sin(โ€“x) = โ€“ sin x)

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.