We know that cosechx=1sinhx
Hence ddxcosechx=ddx1sinhx
Differentiate the given equation with respect to x using quotient rule
ddx1sinhx=[sinhx(d/dx(1))−(1)(d/dx(sinhx)]sinh2x=0−coshxsinh2x=−−coshxsinh2x=−−coshxsinhxsinhx=−coshxsinhx1sinhx=−cothxcosechx
Hence proved that ddxcosechx=−cothxcosechx