Misc 4 - Chapter 3 Class 11 Trigonometric Functions (Important Question)

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Misc 4 Prove that: (cos x – cos y)2 + (sin x – sin y)2 = 4 sin2 (x − y)/2 Solving 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 We know that sin2 x + cos2 x = 1 Replace x by ((𝑥 + 𝑦)/2) sin2 ((x + y)/2) + cos2 ((x + y)/2) = 1