Proove : sinx+sin2x+sin4x+sin5x=4cos x/2.cos3x/2.sin3x
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sinx+sin2x+sin4x+sin5x
=2sin(x+2x)/2cos(x-2x)/2+2sin(4x+5x)/2xos(4x-5x)/2
[sinC+sinD=2sin(C+D)/2cos(C-D)/2]
=2sin3x/2cosx/2+2sin9x/2cosx/2 [∵, cos(-x)=cosx]
=2cosx/2(sin3x/2+sin9x/2)
=2cosx/2{2sin(3x/2+9x/2)/2cos(3x/2-9x/2)/2}
=2cosx/2(2sin12x/4cos6x/4)
=4cosx/2sin3xcos3x/2
=4cosx/2cos3x/2sin3x (Proved)
=2sin(x+2x)/2cos(x-2x)/2+2sin(4x+5x)/2xos(4x-5x)/2
[sinC+sinD=2sin(C+D)/2cos(C-D)/2]
=2sin3x/2cosx/2+2sin9x/2cosx/2 [∵, cos(-x)=cosx]
=2cosx/2(sin3x/2+sin9x/2)
=2cosx/2{2sin(3x/2+9x/2)/2cos(3x/2-9x/2)/2}
=2cosx/2(2sin12x/4cos6x/4)
=4cosx/2sin3xcos3x/2
=4cosx/2cos3x/2sin3x (Proved)
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the last 3rd step was openly shown by pencil at the down portion... Hope it helped you...
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