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Last updated at Feb. 13, 2020 by Teachoo
Maths Crash Course - Live lectures + all videos + Real time Doubt solving!
Example 17 Prove that sin〖5x − 〖2sin 3x +〗sinx 〗/𝑐𝑜𝑠〖5x − 𝑐𝑜𝑠x 〗 = tan x Taking L.H.S. sin〖5x + 〖sin x − 〗2sin3x 〗/𝑐𝑜𝑠〖5x − 𝑐𝑜𝑠x 〗 = 〖(sin〗〖5x + 〖sin x) − 〗〖2 sin〗3x 〗/𝑐𝑜𝑠〖5x − 𝑐𝑜𝑠x 〗 Solving numerator and denominator separately sin 5x + sin x = 2 sin ((5𝑥 + 𝑥)/2) cos ((5𝑥 − 𝑥)/2) = 2 sin (6𝑥/2) cos (4𝑥/2) = 2 sin 3x cos 2x sin 5x + sin x = 2 sin ((5𝑥 + 𝑥)/2) cos ((5𝑥 − 𝑥)/2) = 2 sin (6𝑥/2) cos (4𝑥/2) = 2 sin 3x cos 2x sin x + sin y = 2 sin (𝑥 + 𝑦)/2 cos (𝑥 − 𝑦)/2 Putting x = 5x & y = x sin x + sin y = 2 sin (𝑥 + 𝑦)/2 cos (𝑥 − 𝑦)/2 Putting x = 5x & y = x cos x – cos y = –2 sin (𝑥 + 𝑦)/2 sin (𝑥 − 𝑦)/2 Putting x = 5x & y = x Solving R.H.S 𝐬𝐢𝐧〖𝟓𝐱 + 〖𝐬𝐢𝐧 𝐱 − 〗2sin3x 〗/𝒄𝒐𝒔〖𝟓𝐱 − 𝒄𝒐𝒔𝐱 〗 Putting values = (2 sin3𝑥 cos2𝑥 − 2 sin3𝑥)/(−2 sin〖3𝑥 sin2𝑥 〗 ) = (2 sin3𝑥 (cos〖2𝑥 − 1)〗)/(−2 sin〖3𝑥 sin2𝑥 〗 ) = ( (cos〖2𝑥 − 1)〗)/(−sin2𝑥 ) = ( −(cos〖2𝑥 −1) 〗)/sin2𝑥 = (〖1 − 𝐜𝐨𝐬〗𝟐𝒙 )/𝒔𝒊𝒏𝟐𝒙 "Using cos 2x = 1 – 2sin2 x" "& sin 2x = 2 cos x sin x" = (1 − (𝟏 − 𝟐 𝐬𝐢𝐧𝟐𝒙 ) )/(𝟐 𝒄𝒐𝒔𝒙 𝒔𝒊𝒏𝒙 ) = (1 − 1 + 2 sin2𝑥)/(2 cos〖𝑥 〗 sin𝑥 ) = (0 + 2 sin2𝑥)/(2 cos〖𝑥 〗 sin𝑥 ) = (2 sin2𝑥)/(2 cos〖𝑥 〗 sin𝑥 ) = sin〖𝑥 〗/cos〖𝑥 〗 = tan x = R.H.S. Hence L.H.S. = R.H.S. Hence proved