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Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


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Example 19 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 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) = − "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

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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.