Tangential Identities 

To the first order, the expressions (1+x)/(1−x) and e^{2x} both equal 1+2x, although they differ at higher orders. This suggests that an interesting function might be definable in terms of these expressions. Indeed, the hyperbolic tangent can be defined implicitly by the relation 



Solving for tanh(θ) yields the more usual definition 



This is just the hyperbolic sine divided by the hyperbolic cosine. Putting t = tanh(θ) we have θ = inv tanh(t), and taking the natural log of both sides of equation (1) and dividing through by 2 we get the identity 



Several other useful identities follow from these expressions. For example, given any three quantities θ_{1}, θ_{2}, θ_{3}, we see from equation (1) that 



Taking the natural log of both sides and dividing through by 2, we have 



where t_{j} denotes tanh(θ_{j}). If θ_{1} + θ_{2} + θ_{3} = 0 then the right side of equation (4) equals 1, and we can clear the denominator and cancel terms to arrive at 



More generally, given any n quantities θ_{1}, θ_{2}, ..., θ_{n} and the corresponding values t_{j} = tanh(θ_{j}), we have 



Consequently, we can clear the denominator and rearrange terms to give the identity 



where σ_{m} is the mth elementary symmetric function of the quantities t_{j}. For example, with n = 3 the elementary symmetric functions are 



We stipulate that σ_{m} = 0 for m > n. In terms of these quantities we have 



In other words, the hyperbolic tangent of any sum of quantities equals the sum of the odd elementary symmetric functions of the hyperbolic tangents of those individual quantities divided by the sum of the even elementary symmetric functions. If we consider the n individual quantities as orthogonal coordinates, it’s interesting that this complicated function of the n coordinates in invariant throughout any (n1)dimensional subspace with a given sum, such as the diagonal planes in 3dimensional space. 

With n = 2 we have just two quantities θ_{1}, θ_{2} and the corresponding quantities t_{j} = tanh(θ_{j}), from which we get σ_{0} = 1, σ_{1} = t_{1} + t_{2}, and σ_{2} = t_{1}t_{2}. The above relation then gives the familiar addition rule for the hyperbolic tangent 



Analogously to equation (1), the usual tangent function is defined implicitly by the relation 



Solving for tan(θ) we get the more usual definition 



Putting t = tan(θ), we have θ = inv tan(t), and we can solve equation (5) for θ to give the identity 



Given any three quantities θ_{1}, θ_{2}, θ_{3}, we see from equation (4) that 



Taking the natural log of both sides and dividing through by 2i, we have 



where t_{j} denotes tanh(θ_{j}). If θ_{1} + θ_{2} + θ_{3} = kπ for any integer k, then the right side of (8) equals 1, and we can clear the denominator and cancel terms to arrive at 



where t_{j} denotes tan(θ_{j}). On the other hand, if θ_{1} + θ_{2} + θ_{3} = (k + 1/2)π for any integer k, then the right side of the relation equals −1, and we get 



More generally, given any n quantities θ_{1}, θ_{2}, ..., θ_{n} and the corresponding values t_{j} = tan(θ_{j}), we have 



Consequently, we can clear the denominator and rearrange terms to give the identity 



where σ_{m} is the mth elementary symmetric function of the quantities t_{j}. In terms of these quantities we have 



This can also be written as 



which shows that if s = kπ for any integer k, then the sum of the odd symmetric functions with alternating signs vanishes, whereas if s = (k + 1/2)π then the sum of the even symmetric functions with alternating signs vanishes. 

Now consider three arbitrary quantities q_{1}, q_{2}, q_{3} and let u_{ij} = q_{i} – q_{j}. It follows that 



Putting θ_{1} = u_{12}, θ_{2} = u_{23}, θ_{3} = u_{31}, and letting v_{ij} = tanh(u_{ij}) we see from equation (4) that we have the relation 



Identifying the quantities v_{ij} as the speed of the spatial origin of inertial coordinate system i in terms of inertial coordinate system j, the quantities u_{ij} are called the rapidities, and the above expression gives the rule for velocity composition in special relativity. 
