Setting x=8y, we have to prove that,
cosec2y+cosec4y+cosec8y=coty-cot8y.
Observe that, cosec8y+cot8y=1/(sin8y)+(cos8y)/(sin8y),
=(1+cos8y)/(sin8y),
=(2cos^2 4y)/(2sin4ycos4y),
=(cos4y)/(sin4y).
"Thus, "cosec8y+co8y=cot4y [=cot(1/2*8y)]........(star).
Adding, cosec4y,
cosec4y+(cosec8y+co8y)=cosec4y+cot4y,
=cot(1/2*4y).........[because, (star)].
:. cosec4y+cosec8y+co8y=cot2y.
Re-adding cosec2y and re-using (star),
cosec2y+(cosec4y+cosec8y+co8y)=cosec2y+cot2y,
=cot(1/2*2y).
:.cosec2y+cosec4y+cosec8y+co8y=coty, i.e.,
cosec2y+cosec4y+cosec8y=coty-cot8y, as desired!