((1-cos4x)/sin4x)' - производная

y=((1-cos4x)/sin4x)  y'=(u/v)'=1/v²[u'v-v'u]
u=1-cos4x    u'=4sin4x
v=sin4x  v'=4cos4x
((1-cos4x)/sin4x)'=1/sin²4x[4sin4x*sin4x-4cos4x*(1-cos4x)] =
=4/sin²4x[sin²4x+cos²4x-cos4x]= 4(1-cos4x)/sin²4x

y`=[(1-cos4x)`*sin4x-(sin4x)`(1-cos4x)]/sin²4x=
=(4sin4x*sin4x-4cos4x+4cos²4x)/sin²4x=[4(sin²4x+cos²4x)-4cos4x]/sin²4x=
=(4-4cos4x)/sin²4x=4(1-cos4x)/sin²4x=2sin²2x/(4sin²2xcos²2x)=
=1/(2cos²2x)