Check sibling questions

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 is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.