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

Made by

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.