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Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


Transcript

Example 17 Prove that sin⁡〖5x − 〖2sin 3x +〗⁡sin⁡x 〗/𝑐𝑜𝑠⁡〖5x − 𝑐𝑜𝑠⁡x 〗 = tan x Solving L.H.S. sin⁡〖5x + 〖sin x − 〗⁡2sin⁡3x 〗/𝑐𝑜𝑠⁡〖5x − 𝑐𝑜𝑠⁡x 〗 = 〖(𝐬𝐢𝐧〗⁡〖𝟓𝐱 + 〖𝐬𝐢𝐧 𝐱) − 〗⁡〖𝟐 𝐬𝐢𝐧〗⁡𝟑𝐱 〗/𝒄𝒐𝒔⁡〖𝟓𝐱 − 𝒄𝒐𝒔⁡𝐱 〗 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 cos 5x – cos x = – 2 sin ((𝟓𝒙 + 𝒙)/𝟐) sin((𝟓𝒙 − 𝒙)/𝟐) = – 2 sin (6𝑥/2) sin (4𝑥/2) = – 2 sin 3x sin 2x Solving L.H.S 𝐬𝐢𝐧⁡〖𝟓𝐱 + 〖𝐬𝐢𝐧 𝐱 − 〗⁡2sin⁡3x 〗/𝒄𝒐𝒔⁡〖𝟓𝐱 − 𝒄𝒐𝒔⁡𝐱 〗 Putting values = (𝟐 𝒔𝒊𝒏⁡𝟑𝒙 𝐜𝐨𝐬⁡𝟐𝒙 − 𝟐 𝐬𝐢𝐧⁡𝟑𝒙)/(−𝟐 𝐬𝐢𝐧⁡〖𝟑𝒙 𝒔𝒊𝒏⁡𝟐𝒙 〗 ) = (2 sin⁡3𝑥 (cos⁡〖2𝑥 − 1)〗)/(−2 sin⁡〖3𝑥 sin⁡2𝑥 〗 ) = ( (cos⁡〖2𝑥 − 1)〗)/(−sin⁡2𝑥 ) = ( −(cos⁡〖2𝑥 −1) 〗)/sin⁡2𝑥 = (〖1 − 𝐜𝐨𝐬〗⁡𝟐𝒙 )/𝒔𝒊𝒏⁡𝟐𝒙 = (1 − (𝟏 − 𝟐 𝐬𝐢𝐧𝟐⁡𝒙 ) )/(𝟐 𝒄𝒐𝒔⁡𝒙 𝒔𝒊𝒏⁡𝒙 ) = (1 − 1 + 2 sin2⁡𝑥)/(2 cos⁡〖𝑥 〗 sin⁡𝑥 ) = (0 + 2 sin2⁡𝑥)/(2 cos⁡〖𝑥 〗 sin⁡𝑥 ) = (𝟐 𝐬𝐢𝐧𝟐⁡𝒙)/(𝟐 𝒄𝒐𝒔⁡〖𝒙 〗 𝒔𝒊𝒏⁡𝒙 ) = sin⁡〖𝑥 〗/cos⁡〖𝑥 〗 = tan x = 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.