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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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Transcript

Example 26 Prove that cos 2x cos 𝑥/2 – cos 3x cos 9𝑥/2 = sin 5x sin 5𝑥/2 Solving L.H.S Solving cos 2x cos x/2 and cos 3x cos 9𝑥/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 𝑥/2 = 1/2 ("cos " ("2x + " x/2)" + cos" ("2x" −x/2)) = 1/2 ("cos " ((4x + x )/2)" + cos " ((4x − x)/2)) = 1/2 ("cos " (5x/2)" + cos " (3x/2)) cos 3x cos 𝟗𝒙/𝟐 Replacing x with 3x and y with 9𝑥/2 = 1/2 ("cos" ("3x + " 9𝑥/2)" + cos " ("3x – " 9𝑥/2)) = 1/2 ("cos " ((6𝑥 + 9𝑥 )/2)" + cos " ((6𝑥 − 9𝑥)/2)) = 1/2 ("cos " (15𝑥/2)" + cos " ((−3𝑥)/2)) Now, cos 2x cos 𝑥/2 – cos 3x cos 9𝑥/2 Putting values = 1/2 ("cos " (5x/2)" + cos " (3x/2)) – 1/2 ("cos " (15𝑥/2)" + cos " ((−3𝑥)/2)) = 1/2 ("cos " (5x/2)" + cos " (3x/2)) – 1/2 ("cos " (15𝑥/2)" + cos " (3𝑥/2)) = 1/2 ("cos " (5x/2)" + cos " (3x/2)"– cos " (15𝑥/2)"– cos " (3𝑥/2)) = 1/2 ("cos " (5x/2)"– cos " (15𝑥/2)"+ cos " (3x/2)"– cos " (3𝑥/2)) = 1/2 ("cos " (5x/2)"– cos " (15𝑥/2)"+ " 0) = 1/2 ("cos " (5x/2)"– cos " (15𝑥/2)) = 1/2 ("– 2 sin " ((5𝑥/2 +15/2 𝑥)/2)". sin " ((5𝑥/2 − 15/2 𝑥)/2)) = −"sin " ((5𝑥 + 15𝑥)/(2 × 2))" . sin " ((5𝑥 − 15𝑥)/(2 × 2)) = "– sin " (20𝑥/4)" . sin " (( −10𝑥)/4) Using cos x – cos y = –2 sin (𝑥 + 𝑦)/2 sin (𝑥 − 𝑦)/2 Putting x = 5𝑥/2 , y = 15𝑥/2 = − "sin " (5𝑥)" . sin " (( −5𝑥)/2) = − "sin " (5𝑥) × −" sin " (( 5𝑥)/2) = sin (5𝑥) sin (( 5𝑥)/2) = R.H.S Hence L.H.S. = R.H.S. Hence proved (As sin(–x) = – sin x)

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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 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.