tanh(x)=sinh(x)/cosh(x); csch(x)=1/sinh(x).
sinh(x)=½(ex-e-x), cosh(x)=½(ex+e-x), tanh(x)=(ex-e-x)/(ex+e-x).
The proposed identity is false:
If x=ln(1+√2), sinh(x)=1, cosh(x)=√2, tanh(x)=1/√2.
Therefore tanh2(x)=½, csch2(x)=1, and tanh2(x)-csch2(x)=-½, whereas the proposed identity asserts the answer to be 1.
cosh2(x)-sinh2(x)=¼(e2x+e-2x+2-e2x-e-2x+2)=1.
Therefore, dividing through by sinh2(x):
coth2(x)-1=1/sinh2(x)=csch2(x) and coth2(x)-csch2(x)=1. Using x=ln(1+√2), coth2(x)=1/tanh2(x)=2, and coth2(x)-csch2(x)=2-1=1.