In the previous sections of this study, formulas for various combinations of vector sine and cosine were derived. Based on this methodology, we proceed to consider similar combinations, but in the context of hyperbolic functions [1].

Let us consider in more detail the vector hyperbolic functions of cosine and sine, using the transformation \[ \ch\v \to \Ch\v \\ \sh\v \to \Sh\v, \tag{1}\] presented in this source. Let us write the corresponding formulas in the following form: \[ \Ch\v = \j_0 + \sumn{1} \j_n \sqrt{ {1 + (-1)^n \over 4} {\a^{n} \over n!} } \\ \Sh\v = \sumn{1} \j_n \sqrt{ {1 + (-1)^n \over 4} {\a^{n} \over n!} } \tag{2}\] \[ \v = {\a \over 2} \tag{3}\] where \(\v\) is the angle taken for convenience of display as half of \(\a\).

Based on the main property of transforming a scalar into a vector, we denote that: \[ [\Ch\v]^2 = \ch^2 \v \\ [\Sh\v]^2 = \sh^2 \v \tag{4}\] From here, using the properties of decreasing the degree of hyperbolic functions [1], we can immediately derive the formula for the sum of squares: \[ [\Ch\v]^2 + [\Sh\v]^2 = \ch\a \tag{5}\] For subsequent actions, we will need the product of the hyperbolic cosine and sine \[ \Ch\v \cdot \Sh\v = \sumn{1} {1 + (-1)^n \over 4} {\a^{n} \over n!} = \frac12 \sumn{0} {\a^{2n} \over (2n)!} - \frac12, \tag{6}\] where, applying the sum of the Maclaurin series [2], we get: \[ \Ch\v \cdot \Sh\v = {\ch\a - 1 \over 2} = \sh^2 \v \tag{7}\] Now it is quite easy to find the square of the sum of the hyperbolic cosine and sine \[ [\Ch\v + \Sh\v]^2 = 2\, \ch\a - 1 = 1 + 4\, \sh^2 \v, \tag{8}\] and square of the difference \[ [\Ch\v - \Sh\v]^2 = 1 \tag{9}\] But the difference between the hyperbolic cosine and sine will be equal to the zeroth unit vector: \[ \Ch\v - \Sh\v = \pm \mathbf{j_0} \tag{10}\] This relationship follows directly from formula (2).

Of particular interest may be the products of vector hyperbolic functions with two angles: \[ \v_1 = {\a_1 \over 2}, \quad \v_2 = {\a_2 \over 2} \tag{11}\] Using transformation (2) we can find the product of two hyperbolic sines with different angles \[ \Sh\v_1 \cdot \Sh\v_2 = \sumn{1} {1 + (-1)^n \over 4} {\sqrt{\mathstrut \a_1^{n} \a_2^{n}} \over n!} = \frac12 \sumn{1} {(\a_1 \a_2)^{n} \over (2n)!} \tag{12}\] Obviously, the sum of the series is equal to the scalar hyper cosine without unity, which follows from the Maclaurin series [2], if we assume that \[ \sqrt{\a_1 \a_2} = x \] Then \[ \Sh\v_1 \cdot \Sh\v_2 = \frac12 \sumn{0} {x^{2n} \over (2n)!} - \frac12 = { \ch x - 1 \over 2} = \sh^2 {x \over 2} \tag{13}\] Substituting the initial angles for \(x\) we obtain: \[ \Sh\v_1 \cdot \Sh\v_2 = \sh^2 \vvq \tag{14}\] The product of two vector hypercosines with different angles is proved in a similar way: \[ \Ch\v_1 \cdot \Ch\v_2 = \ch^2 \vvq \tag{15}\] For vector hyperbolic functions, we can also construct a beautiful and completely non-obvious expression that follows from our calculations: \[ \Ch\v_1 \cdot \Ch\v_2 - \Sh\v_1 \cdot \Sh\v_2 = 1 \tag{16}\] Compare it with formula (10) from the previous section.

Let's combine the formulas obtained from this part of the work into one table.

No.	Action	Result
1	\( \Sh\v_1 \cdot \Sh\v_2 \)	\( \sh^2 \vvq \)
2	\( \Ch\v_1 \cdot \Ch\v_2 \)	\( \ch^2 \vvq \)
3	\( \Ch\v_1 \cdot \Ch\v_2 - \Sh\v_1 \cdot \Sh\v_2 = 1 \)
4	\( \Ch\v_1 \cdot \Sh\v_2 \)	\( \sh^2 \vvq \)
5	\( \left[ \Ch\v + \Sh\v \right]^2 \)	\( 1 + 4\, \sh^2\v \)
6	\( \left[ \Ch\v - \Sh\v \right]^2 \)	\( 1 \)
7	\( \Ch\v - \Sh\v \)	\( \pm \mathbf{j_0} \)
8	\( |\Ch\v| \)	\( \ch\v \)
9	\( |\Sh\v| \)	\( \pm\, \sh\v \)
10	\( |\Ch\v + \Sh\v|^2 = [\Ch\v]^2 + [\Sh\v]^2 = \ch\a \)

This angle can be derived quite easily from the formulas obtained earlier: \[ \cos \theta = {\Ch\v \cdot \Sh\v \over |\Ch\v| |\Sh\v| } = \pm \mathbb{th}\, \v \tag{17}\] Thus, the cosine of the angle between the hyperbolic sine and cosine vectors is equal to the hyperbolic tangent, from which the angle itself can be found. It can be shown that if \(\v\) tends to a large value, the angle between the vectors tends to zero.

Main results of the work:

• It is established that vector hyperbolic functions retain their connection with classical scalar functions.
• The properties of their squares, sums and differences are derived.
• It is shown that the products and combinations of these functions form strict patterns similar to the known identities for ordinary hyperbolic functions.
• Non-trivial results for functions with different angles are found, which lead to unexpectedly simple and beautiful relationships.
• All the obtained conclusions are summarized in a table for ease of use.

In general, the work extends the classical theory of hyperbolic functions to the vector case, preserving its internal logic and revealing new symmetric patterns.