Misc 4 - Prove (cos x - cos y)2 + (sin x - sin y)2 - Chapter 3 - cos x + cos y formula

  1. Class 11
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Misc 4 Prove that: (cos x – cos y)2 + (sin x – sin y)2 = 4 sin2 (x − y)/2 Taking L.H.S (cos x – cos y )2 + (sin x – sin y )2 = (−"2 sin (" (𝑥 + 𝑦)/2 ") sin (" (𝑥 − 𝑦)/2 ")" )^2 + ("2 cos (" (𝑥 + 𝑦)/2 ") sin (" (𝑥 − 𝑦)/2 ")" )^2 = ((−"2)2 sin2 (" (𝑥 + 𝑦)/2 ") sin2 (" (𝑥 − 𝑦)/2 ")" ) + ("22 cos2 (" (𝑥 + 𝑦)/2 ") sin2 (" (𝑥 − 𝑦)/2 ")" ) = 4 sin2 ((𝑥 + 𝑦)/2) . sin2 ((𝑥 − 𝑦)/2) + 4 cos2 ((𝑥 + 𝑦)/2) sin2 ((𝑥 − 𝑦)/2) = 4 sin2 ( (𝑥 − 𝑦)/2) ("sin2 ( " (𝑥 + 𝑦)/2 ") + cos2 ( " (𝑥+ 𝑦)/2 ")" ) = 4 sin2 ( (𝑥 − 𝑦)/2) × 1 = 4 sin2 ( (𝑥 − 𝑦)/2) = R.H.S. Hence , L.H.S.= R.H.S. Hence proved

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