Ex 3.3, 19 - Chapter 3 Class 11 Trigonometric Functions

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Ex 3.3, 19 Prove that 〖sin x +〗⁡sin⁡3x /(𝑐𝑜𝑠⁡x + 𝑐𝑜𝑠⁡3x ) = tan 2x Solving L.H.S. 〖sin x +〗⁡sin⁡3x /(𝑐𝑜𝑠⁡x + 𝑐𝑜𝑠⁡3x ) We solve sin x + sin 3x & cos x + cos 3x seperately sin x + sin 3x = 2 sin ((x+3x)/2) cos ((x−3x)/2) = 2 sin (4𝑥/2) cos ((−2𝑥)/2) = 2 sin 2x cos (–x) cos x + cos 3x = 2 cos ((x+3x)/2) cos ((5x−3x)/2) = 2 cos (4𝑥/2) cos ((−2𝑥)/2) = 2 cos 2x cos (–x) Now 𝑠𝑖𝑛⁡〖𝑥 + 𝑠𝑖𝑛⁡3𝑥 〗/𝑐𝑜𝑠⁡〖𝑥 + 𝑐𝑜𝑠⁡3𝑥 〗 = (𝟐 〖 𝒔𝒊𝒏 〗⁡〖𝟐𝒙 𝒄𝒐𝒔⁡〖(−𝒙)〗 〗)/(𝟐 𝒄𝒐𝒔⁡〖 𝟐𝒙 𝒄𝒐𝒔⁡〖(−𝒙)〗 〗 ) = 𝑠𝑖𝑛⁡〖 2x〗/cos⁡〖 2x〗 = tan 2x = R.H.S Hence L.H.S = R.H.S Hence proved