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

Misc 4 - Chapter 3 Class 11 Trigonometric Functions - Part 2

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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) We know that cos x – cos y = –2 sin ((𝑥 + 𝑦)/2) sin ((𝑥 − 𝑦)/2) sin x – sin y = 2cos ((𝑥 + 𝑦)/2) sin ((𝑥 − 𝑦)/2) We know that sin2 x + cos2 x = 1 Replace x by ((𝑥 + 𝑦)/2) sin2 ((x + y)/2) + cos2 ((x + y)/2) = 1 = 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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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.