Last updated at Feb. 13, 2020 by Teachoo

Transcript

Misc 2 Prove that: (sin 3๐ฅ + sin ๐ฅ) sin ๐ฅ + (cos 3๐ฅ โ cos ๐ฅ) cos ๐ฅ = 0 Lets calculate (sin 3x + sin x) and (cos 3x โ cos x) separately We know that sin x + sin y = sin ((๐ฅ + ๐ฆ)/2) cos ((๐ฅ โ ๐ฆ)/2) Replacing x with 3x and y with x sin 3x + sin x = 2sin ((3๐ฅ + ๐ฅ)/2) cos ((3๐ฅ โ ๐ฅ)/2) sin 3x + sin x = 2 sin 2x cos x Similarly , We know that cos x โ cos y = โ2 sin ((๐ฅ + ๐ฆ)/2) sin ((๐ฅ โ ๐ฆ)/2) Replacing x with 3x and y with x cos 3x โ cos x = โ2 sin ((3๐ฅ + ๐ฅ)/2) sin ((3๐ฅ โ ๐ฅ)/2) cos 3x โ cos x = โ2 sin 2x sin x Now solving L.H.S (sin 3x + sin x) sin x + (cos 3x โ cos x) cos x Putting values from (1) & (2) = (2 sin 2x cos x) (sin x) + (โ2sin 2x) (sin x) (cos x) = 2 sin 2x cos x sin x โ 2 sin 2x sin x cos x = 0 = R.H.S Hence L.H.S = R.H.S Hence proved

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.