Misc 2 - Prove (sin 3x + sin x) sin x + (cos 3x - cos x) - 2x 3x formula - Proving

  1. Chapter 3 Class 11 Trigonometric Functions
  2. Serial order wise
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Misc, 2 Prove that: (sin 3 + sin ) sin + (cos 3 cos ) cos = 0 Lets calculate (sin 3x + sin x) and (cos 3x cos x) separately We know that sin x + sin y = sin (( + )/2) cos (( )/2) Replacing x with 3x and y with x Hence sin 3x + sin x = 2sin ((3 + )/2) cos ((3 )/2) sin 3x + sin x = 2 sin 2x cos x Similarly , we know that cos x cos y = 2 sin (( + )/2) sin (( )/2) Replacing x with 3x and y with x Hence cos 3x cos x = 2 sin ((3 + )/2) sin ((3 )/2) cos 3x cos x = 2 sin 2x sin x Now solving L.H.S (sin 3x + sin x) sin x + (cos 3x cos x) cos x From (1) & (2) = (2 sin 2x cos x) (sin x) + ( 2sin 2x) (sin x) (cos x) = 2 sin 2x cos x sin x 2 sin 2x sin x cos x = 0 =R.H.S Hence L.H.S = R.H.S Hence proved

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.