Prove that
ranitkarmaka:
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Let theta = x
LHS sin5x
=sin(3x+2x)
=sin3x.cos2x+cos3x.sin2x
=(3sinx-4sin^3x).(1–2sin^2x)+(4cos^3x-3cosx).(2sinx.cosx)
=(3sinx-4sin^3x)(1–2sin^2x)+(4cos^4x-3cos^2x)(2sinx).
=(3sinx-4sin^3x)(1–2sin^2x)+cos^2x.(4cos^2x-3).(2sinx)
=3sinx-4sin^3x-6sin^3x+8sin^5x+(1-sin^2x).(4–4sin^2x-3).(2sinx)
=8sin^5x-10sin^3x+3sinx+(2sinx-2sin^3x)(1–4sin^2x).
=8sin^5x-10sin^3x+3sinx+2sinx-2sin^3x-8sin^3x+8sin^5x
=16sin^5x-20sin^3x+5sinx proved
LHS sin5x
=sin(3x+2x)
=sin3x.cos2x+cos3x.sin2x
=(3sinx-4sin^3x).(1–2sin^2x)+(4cos^3x-3cosx).(2sinx.cosx)
=(3sinx-4sin^3x)(1–2sin^2x)+(4cos^4x-3cos^2x)(2sinx).
=(3sinx-4sin^3x)(1–2sin^2x)+cos^2x.(4cos^2x-3).(2sinx)
=3sinx-4sin^3x-6sin^3x+8sin^5x+(1-sin^2x).(4–4sin^2x-3).(2sinx)
=8sin^5x-10sin^3x+3sinx+(2sinx-2sin^3x)(1–4sin^2x).
=8sin^5x-10sin^3x+3sinx+2sinx-2sin^3x-8sin^3x+8sin^5x
=16sin^5x-20sin^3x+5sinx proved
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Let theta = x
LHS sin5x
=sin(3x+2x)
=sin3x.cos2x+cos3x.sin2x
=(3sinx-4sin^3x).(1–2sin^2x)+(4cos^3x-3cosx).(2sinx.cosx)
=(3sinx-4sin^3x)(1–2sin^2x)+(4cos^4x-3cos^2x)(2sinx).
=(3sinx-4sin^3x)(1–2sin^2x)+cos^2x.(4cos^2x-3).(2sinx)
=3sinx-4sin^3x-6sin^3x+8sin^5x+(1-sin^2x).(4–4sin^2x-3).(2sinx)
=8sin^5x-10sin^3x+3sinx+(2sinx-2sin^3x)(1–4sin^2x).
=8sin^5x-10sin^3x+3sinx+2sinx-2sin^3x-8sin^3x+8sin^5x
=16sin^5x-20sin^3x+5sinx proved
LHS sin5x
=sin(3x+2x)
=sin3x.cos2x+cos3x.sin2x
=(3sinx-4sin^3x).(1–2sin^2x)+(4cos^3x-3cosx).(2sinx.cosx)
=(3sinx-4sin^3x)(1–2sin^2x)+(4cos^4x-3cos^2x)(2sinx).
=(3sinx-4sin^3x)(1–2sin^2x)+cos^2x.(4cos^2x-3).(2sinx)
=3sinx-4sin^3x-6sin^3x+8sin^5x+(1-sin^2x).(4–4sin^2x-3).(2sinx)
=8sin^5x-10sin^3x+3sinx+(2sinx-2sin^3x)(1–4sin^2x).
=8sin^5x-10sin^3x+3sinx+2sinx-2sin^3x-8sin^3x+8sin^5x
=16sin^5x-20sin^3x+5sinx proved
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