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Find sum till n terms of (Sinx/cos3x) + (sin3x/cos9x) +(sin9x/cos27x)+(sin27x/cos81x)+......

Find sum till n terms of
(Sinx/cos3x) + (sin3x/cos9x) +(sin9x/cos27x)+(sin27x/cos81x)+......

Grade:11

1 Answers

Akshay
185 Points
8 years ago
Sum = ∑ sin(3jx)/cos(3j+1x), multiply and divide by cos(3jx),
Sum = ∑ sin(2*3jx) / 2.cos(3jx).cos(3j+1x)....using sin2x=2sinx.cosx,
Sum = ∑ sin((3-1)*3jx) / 2.cos(3jx).cos(3j+1x),
Sum = ∑ [sin(3j+1x).cos(3jx) – cos(3j+1x).sin(3jx)] / 2.cos(3jx).cos(3j+1x) ,
Sum = ½ ∑ tan(3j+1x) – tan(3jx) = ½ * (tan3x – tanx + tan9x – tan3x + tan81x – tan9x......)
terms will cancel each other,
Sum = ½ * (tan(3n+1x) – tanx)

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