This topic, started in the previous part, will be incomplete if we do not try to multiply vector sines and cosines with different angles. In doing so, we will obtain completely unexpected results that are fundamentally unattainable with scalar multiplication of trigonometric functions. For this, we will apply the apparatus of hyperbolic vector algebra, Maclaurin series [1] and classical trigonometry.

Let us recall the formulas for vector cosine and sine: \[ \S = \sumn{1} \j_{2n}\, (\i)^n \sqrt{ \frac12 {\a^{2n} \over (2n)!} } \\ \S^{*} = \sumn{1} \j_{2n}\, (\oi)^n \sqrt{ \frac12 {\a^{2n} \over (2n)!} } \tag{1}\] \[ \Cos\v = \j_0 + \S \tag{2}\] \[ \Sin\v = \ik \S \tag{3}\] Here: \(\j_n\) - unit vectors of a multidimensional space, \(\i\) - hyperbolic unit whose square is plus one, and \(\oi\) - the unit complex conjugate to it, \(\v\) - the angle taken for convenience of display as half of \(\a\). The vector sine and cosine are denoted respectively: \(\Sin\v\) and \(\Cos\v\).

But in this part of the work we consider the action with two different angles, so we denote \[ \v_1 = {\a_1 \over 2}, \quad \v_2 = {\a_2 \over 2} \tag{4}\] First we find the product of the vector sines of the two angles using (3) \[ \Sin\v_1 \cdot \Sin\v_2 = \oi \S_1^{*} \cdot \ik \S_2 = \oi \ik \sumn{1} (\oi)^n (\i)^n \frac12 { (\aaq)^{2n} \over (2n)!} \tag{5}\] Since in hyperbolic algebra \[ \oi\kern1pt^n \cdot \i^n = (-1)^n \] then \[ \Sin\v_1 \cdot \Sin\v_2 = -\sumn{1} (-1)^n \frac12 {(\aaq)^{2n} \over (2n)!} \tag{6}\] Obviously, the sum of the series is equal to the scalar cosine minus one, which follows from the Maclaurin series [1], if we assume that \[ \sqrt{\mathstrut \a_1 \a_2} = x \] From here we obtain the final expression \[ \Sin\v_1 \cdot \Sin\v_2 = {1 - \cos(\aaq) \over 2} = \sin^2\vvq \tag{7}\] Try to evaluate how significantly — both in form and in content — the multiplication of scalar trigonometric functions and their vector analogues differs.

Let's do the same with the product of vector cosines of two angles \[ \Cos\v_1 \cdot \Cos\v_2 = (\j_0 + \S_1^{*}) \cdot (\j_0 + \S_2) = 1 + \S_1^{*} \cdot \S_2 \tag{8}\] Here the situation is similar, and the sum of the series is equal to the scalar cosine without unity. Substituting it into the previous expression, we get: \[ \Cos\v_1 \cdot \Cos\v_2 = {1 + \cos \sqrt{\mathstrut \a_1 \a_2} \over 2} = \cos^2 \vvq \tag{9}\] From formulas (7) and (9), the following beautiful expression is automatically obtained: \[ \Sin\v_1 \cdot \Sin\v_2 + \Cos\v_1 \cdot \Cos\v_2 = 1 \tag{10}\] This non-obvious property can be applied to vector rotation matrices.

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

No.	Action	Result
1	\( \Sin\v_1 \cdot \Sin\v_2 \)	\( \sin^2 \vvq \)
2	\( \Cos\v_1 \cdot \Cos\v_2 \)	\( \cos^2 \vvq \)
3	\( \Sin\v_1 \cdot \Sin\v_2 + \Cos\v_1 \cdot \Cos\v_2 = 1 \)
4	\( \Cos\v_1 \cdot \Sin\v_2 \)	\( -\ik \sin^2 \vvq \)
5	\( \Sin\v_1 \cdot \Cos\v_2 \)	\( \ik \sin^2 \vvq \)
6	\( \left[ \Cos\v + \Sin\v \right]^2 \)	\( 1 \)
7	\( \left[ \Cos\v - \Sin\v \right]^2 \)	\( 1 \)
8	\( | \Cos\v + \Sin\v |^2 = |\Cos\v|^2 + |\Sin\v|^2 = 1 \)
9	\( |\Cos\v - \ik\Sin\v | \)	\( 1 \)
10	\( \Cos\v - \ik\Sin\v \)	\( \j_0 \)

For clarity of the analysis of the rows in Table 1, we clarify that in quadratic notation, vector factors, in general, do not commute.

This consideration of vector sines and cosines shows that their behavior during multiplication differs significantly from classical trigonometric functions. The obtained regularities not only expand the mathatic understanding of trigonometry, but also open up possibilities for practical applications in the field of matrix transformations and rotation modeling.