Question 2 Prove that: (sin 3x + sin x) sin x + (cos 3x – cos x) cos x = 0
Class X1 - Maths -Trigonometric Functions Page 81
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LHS = (sin3x + sinx )sinx + (cos3x - cosx)cosx
use the formula,
sinC + sinD = 2sin(C+D)/2.cos(C-D)/2
cosC-cosD = 2sin(C+D)/2.sin(D-C)/2
= {2sin(3x+x)/2.cos(3x-x)/2}sinx + {2sin(3x+x)/2.sin(x-3x)/2}cosx
= (2sin2x.cosx)sinx - (2sin2x.sinx)cosx
= 2sin2x.{sinx.cosx - sinx.cosx }
= 0 = RHS
use the formula,
sinC + sinD = 2sin(C+D)/2.cos(C-D)/2
cosC-cosD = 2sin(C+D)/2.sin(D-C)/2
= {2sin(3x+x)/2.cos(3x-x)/2}sinx + {2sin(3x+x)/2.sin(x-3x)/2}cosx
= (2sin2x.cosx)sinx - (2sin2x.sinx)cosx
= 2sin2x.{sinx.cosx - sinx.cosx }
= 0 = RHS
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