Misc 3 - Prove (cos x + cos y)2 + (sin x - sin y)2 - Class 11 - cos x + cos y formula

  1. Chapter 3 Class 11 Trigonometric Functions
  2. Serial order wise
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Misc 3 Prove that: (cos x + cos y)2 + (sin x โ€“ sin y)2 = 4 cos2 (x + y)/2 Taking L.H.S (cos x + cos y )2 + (sin x โ€“ sin y )2 = ("2cos (" (๐‘ฅ + ๐‘ฆ)/2 ") cos (" (๐‘ฅ โˆ’ ๐‘ฆ)/2 ")" )^2 + ("2 cos (" (๐‘ฅ + ๐‘ฆ)/2 ") sin (" (๐‘ฅ โˆ’ ๐‘ฆ)/2 ")" )^2 = ("22 cos2 (" (๐‘ฅ + ๐‘ฆ)/2 ") cos2 (" (๐‘ฅ โˆ’ ๐‘ฆ)/2 ")" ) + ("22 cos2 (" (๐‘ฅ + ๐‘ฆ)/2 ") sin2 (" (๐‘ฅ โˆ’ ๐‘ฆ)/2 ")" ) = 4 cos2 ((๐‘ฅ + ๐‘ฆ)/2) . cos2 ((๐‘ฅ โˆ’ ๐‘ฆ)/2) + 4 cos2 ((๐‘ฅ + ๐‘ฆ)/2) sin2 ((๐‘ฅ โˆ’ ๐‘ฆ)/2) = 4cos2 ( (๐‘ฅ + ๐‘ฆ)/2) ("cos2 ( " (๐‘ฅ โˆ’ ๐‘ฆ)/2 ") + sin2 ( " (๐‘ฅ โˆ’ ๐‘ฆ)/2 ")" ) = 4 cos2 ( (๐‘ฅ + ๐‘ฆ)/2) ร— 1 = 4 cos2 ( (๐‘ฅ + ๐‘ฆ)/2) = R.H.S. Hence , L.H.S.= R.H.S. Hence proved

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