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cosec x- cosec 2x = cosec4x

Consider the following equation.    cosecx-cosec2x=cosec4xUse the reciprocal identity, cosecx=1sinx to get,    1sinx-1sin2x=1sin4x                    1sinx=1sin2x+1sin4x                    1sinx=sin4x+sin2xsin2x sin4x       sin2x sin4xsinx=sin4x+sin2xUse the trigonometric identities.            sin2x=2sinx cosx   sinx+siny=2 sinx+y2cosx-y2This implies that,      2sinx cosx sin4xsinx=2 sin4x+2x2cos4x-2x2                 2cosx sin4x=2 sin3x cosxCancel the common factors on both sides to get,    sin4x=sin3xNow sinx=siny implies that   x=+-1ny,  nZThis gives,                      4x=+-1n3x,  nZ     4x--1n3x=,  nZ    4x+-1-1n3x=,  nZ     4x+3-1n+1x=,  nZ     4+3-1n+1x=,  nZTherefore the required solution is,    x=4+3-1n+1,  nZ

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