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Example 26 - Prove cos 2x cos x/2 - cos 3x cos 9x/2 = sin 5x

Example 26 - Chapter 3 Class 11 Trigonometric Functions - Part 2
Example 26 - Chapter 3 Class 11 Trigonometric Functions - Part 3 Example 26 - Chapter 3 Class 11 Trigonometric Functions - Part 4 Example 26 - Chapter 3 Class 11 Trigonometric Functions - Part 5

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Example 26 Prove that cos 2x cos π‘₯/2 – cos 3x cos 9π‘₯/2 = sin 5x sin 5π‘₯/2 Solving L.H.S Solving cos 2x cos x/2 and cos 3x cos 9π‘₯/2 separately We know that 2 cos x cos y = cos (x + y) + cos (x – y) cos x cos y = 1/2 ("cos (x + y) + cos (x – y)" ) cos 2x cos 𝒙/𝟐 Replacing x with 2x and y with π‘₯/2 = 1/2 ("cos " ("2x + " x/2)" + cos" ("2x" βˆ’x/2)) = 1/2 ("cos " ((4x + x )/2)" + cos " ((4x βˆ’ x)/2)) = 1/2 ("cos " (5x/2)" + cos " (3x/2)) cos 3x cos πŸ—π’™/𝟐 Replacing x with 3x and y with 9π‘₯/2 = 1/2 ("cos" ("3x + " 9π‘₯/2)" + cos " ("3x – " 9π‘₯/2)) = 1/2 ("cos " ((6π‘₯ + 9π‘₯ )/2)" + cos " ((6π‘₯ βˆ’ 9π‘₯)/2)) = 1/2 ("cos " (15π‘₯/2)" + cos " ((βˆ’3π‘₯)/2)) Now, cos 2x cos π‘₯/2 – cos 3x cos 9π‘₯/2 Putting values = 1/2 ("cos " (5x/2)" + cos " (3x/2)) – 1/2 ("cos " (15π‘₯/2)" + cos " ((βˆ’3π‘₯)/2)) = 1/2 ("cos " (5x/2)" + cos " (3x/2)) – 1/2 ("cos " (15π‘₯/2)" + cos " (3π‘₯/2)) = 1/2 ("cos " (5x/2)" + cos " (3x/2)"– cos " (15π‘₯/2)"– cos " (3π‘₯/2)) = 1/2 ("cos " (5x/2)"– cos " (15π‘₯/2)"+ cos " (3x/2)"– cos " (3π‘₯/2)) = 1/2 ("cos " (5x/2)"– cos " (15π‘₯/2)"+ " 0) = 1/2 ("cos " (5x/2)"– cos " (15π‘₯/2)) = 1/2 ("– 2 sin " ((5π‘₯/2 +15/2 π‘₯)/2)". sin " ((5π‘₯/2 βˆ’ 15/2 π‘₯)/2)) = βˆ’"sin " ((5π‘₯ + 15π‘₯)/(2 Γ— 2))" . sin " ((5π‘₯ βˆ’ 15π‘₯)/(2 Γ— 2)) = "– sin " (20π‘₯/4)" . sin " (( βˆ’10π‘₯)/4) Using cos x – cos y = –2 sin (π‘₯ + 𝑦)/2 sin (π‘₯ βˆ’ 𝑦)/2 Putting x = 5π‘₯/2 , y = 15π‘₯/2 = βˆ’ "sin " (5π‘₯)" . sin " (( βˆ’5π‘₯)/2) = βˆ’ "sin " (5π‘₯) Γ— βˆ’" sin " (( 5π‘₯)/2) = sin (5π‘₯) sin (( 5π‘₯)/2) = R.H.S Hence L.H.S. = R.H.S. Hence proved (As sin(–x) = – sin x)

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.