Ex 3.3, 17 - Chapter 3 Class 11 Trigonometric Functions

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Ex 3.3, 17 Prove that 𝑠𝑖𝑛⁡〖5𝑥 + 𝑠𝑖𝑛⁡3𝑥 〗/𝑐𝑜𝑠⁡〖5𝑥 + 𝑐𝑜𝑠⁡3𝑥 〗 = tan 4x Solving L.H.S. 𝑠𝑖𝑛⁡〖5𝑥 + 𝑠𝑖𝑛⁡3𝑥 〗/𝑐𝑜𝑠⁡〖5𝑥 + 𝑐𝑜𝑠⁡3𝑥 〗 We solve sin 5x + sin 3x & cos 5x + cos 3x seperately sin 5x + sin 3x = 2 sin ((5x+3x)/2) cos ((5x−3x)/2) = 2 sin (8𝑥/2) cos (2𝑥/2) = 2 sin 4x cos x cos 5x + cos 3x = 2 cos ((5x+3x)/2) cos ((5x−3x)/2) = 2 cos (8𝑥/2) cos (2𝑥/2) = 2 cos 4x cos (x) Now 𝒔𝒊𝒏⁡〖𝟓𝒙 + 𝒔𝒊𝒏⁡𝟑𝒙 〗/𝒄𝒐𝒔⁡〖𝟓𝒙 + 𝒄𝒐𝒔⁡𝟑𝒙 〗 = (2 〖 𝑠𝑖𝑛 〗⁡〖4x 𝑐𝑜𝑠⁡𝑥 〗)/(2 𝑐𝑜𝑠⁡〖 4𝑥 𝑐𝑜𝑠⁡𝑥 〗 ) = 𝒔𝒊𝒏⁡〖 𝟒𝒙〗/𝐜𝐨𝐬⁡〖 𝟒𝒙〗 = tan 4x = R.H.S Hence L.H.S = R.H.S Hence proved