tanhx = sinhx / coshx(min.value-1 & max.value1) cosh2x - sinh2x = 1 ;        cosh2x + sinh2x = cosh(2x)
cothx = coshx / sinhx sech2x + tanh2x = 1             2sinhx*coshx = sinh(2x)
sechx = 1 / coshx coth2x - csch2x = 1       2tanhx/(1+tanh2x)=tanh(2x)
cschx = 1 / sinhx 1 - tanh2x = 1 / cosh2x   
cothx = 1/ tanhx coth2x - 1 = 1 / sinh2x
sinh(-x) = - sinhx coshx+sinhx=ex
cosh(-x)= coshx coshx-sinhx=e-x
tanh(-x) = -tanhx  
sin x = (eix - e-ix ) / 2i sin x = x - x3 / 3! +x5 / 5! -x7 / 7! +...........
sinhx = (ex - e-x ) / 2 sinhx = 1 + x3 / 3! + x5 / 5! + x7 / 7! +...........
cos x = (eix + e-ix ) / 2 cos x = 1 - x2 / 2! +x4 / 4! -x6 / 6! +...........
coshx = (ex + e-x ) / 2 coshx = 1 + x2 / 2! + x4 / 4! + x6 / 6! +...........(*)
sinhx = -i sin(ix) (*) The equation of parabola can be written as y=1 + x2 / 2! So for small values of x, y=coshx is almost similar to a parabola.
coshx = cos(ix) sinh(x+y) =sinhx*coshy +coshx*sinhy
tanhx=-itanh(ix) cosh(x+y) =coshx*coshy +sinhx*sinhy
cothx=icoth(ix) tanh(xy) =(tanhxtanhy)/(1tanhx*tanhy)
d(sinhx)/dx=coshx  , d(coshx)/dx=sinhx sinhxsinhy=2(sinh(xy)/2)cosh((xy)/2))
d(tanhx)/dx=sech2x   , d(cothx)/dx=-cosech2x coshxcoshy=2(cosh(xy)/2)cosh((xy)/2))
d(sechx)=-sechx*tanhx tanh(x/2) =sinhx/(coshx+1) =(coshx-1)/sinhx
d(cosechx)=-cosechx*cothx  
d2(sinhx)/dx2 =sinhx

All functions with this property are linear combinations of sinh and cosh

Area of a circular section with radius r and angle θ (in radians) is  rθ2 /2. It will be equal to θ when r=√2. Such a circle is tangent to the hyperbola xy=1 at (1,1)
d2(coshx)/dx2 =coshx

All functions with this property are linear combinations of sinh and cosh

The hyperbolic angle is an invariant measure with respect to squeeze mapping just as the circular angle is an invariant measure under rotation. The area of a hyperbolic sector is taken as the measure of the hyperbolic angle associated with the sector. The hyperbolic angle concept is quite independent of the concept of ordinary circular angle but shares a property of invariance with it. Whereas circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping.Both circular and hyperbolic angle generate invariant measures but with respect to different transformation groups. The hyperbolic functions, which take hyperbolic angle as argument, perform the role that circular functions play with the circular angle argument.

   
   
   

 

 x ≥ 0 sinhx = u coshx = u tanhx = u cothx = u sechx = u cschx = u
sinhx u √(u2-1) u / √(1-u2) 1 / √(u2-1) √(1-u2) / u 1 / u
coshx √(1+u2) u 1 / √(1-u2) u / √(u2-1) 1 / u √(1+u2) / u
tanhx u / √(1+u2) √(u2-1) / u u 1 / u √(1-u2) 1 / √(1+u2)
cothx √(1+u2) / u u / √(u2-1) 1 / u u 1 / √(1-u2) √(1+u2)
sechx 1 / √(1+u2) 1/u √(1-u2) √(u2-1) / u u u / √(1+u2)
cschx 1 / u 1 / √(u2-1) √(1-u2) / u √(u2-1) u / √(1-u2) u
x < 0 sinhx = -u coshx = u tanhx = -u cothx = -u sechx = u cschx = - u
sinhx - u √(u2-1) -u / √(1-u2) -1 / √(u2-1) √(1-u2) / u -1 / u
coshx √(1+u2) u 1 / √(1-u2) u / √(u2-1) 1 / u √(1+u2) / u
tanhx -u / √(1+u2) √(u2-1) / u -u -1 / u √(1-u2) -1 / √(1+u2)
cothx -√(1+u2) / u u / √(u2-1) -1 / u -u 1 / √(1-u2) -√(1+u2)
sechx 1 / √(1+u2) 1/u √(1-u2) √(u2-1) / u u u / √(1+u2)
cschx -1 / u 1 / √(u2-1) -√(1-u2) / u -√(u2-1) u / √(1-u2) -u