12 Conversion Formulas.

Article 12
Conversion Formulas.

To prove that

\begin{aligned} cosh u_{1} + cosh u_{2} & = 2 cosh \frac{1}{2} (u_{1} + u_{2}) cosh \frac{1}{2} (u_{1} - u_{2}), \\ cosh u_{1} - cosh u_{2} & = 2 sinh \frac{1}{2} (u_{1} + u_{2}) sinh \frac{1}{2} (u_{1} - u_{2}), \\ sinh u_{1} + sinh u_{2} & = 2 sinh \frac{1}{2} (u_{1} + u_{2}) cosh \frac{1}{2} (u_{1} - u_{2}), \\ sinh u_{1} - sinh u_{2} & = 2 cosh \frac{1}{2} (u_{1} + u_{2}) sinh \frac{1}{2} (u_{1} - u_{2}) . \end{aligned}\}

(23)

From the addition formulas it follows that

\begin{array}{l} cosh (u + v) + cosh (u - v) & = 2 cosh u cosh v, \\ cosh (u + v) - cosh (u - v) & = 2 sinh u sinh v, \\ sinh (u + v) + sinh (u - v) & = 2 sinh u cosh v, \\ sinh (u + v) - sinh (u - v) & = 2 cosh u sinh v, \end{array}

and then by writing $u + v = u_{1}$ , $u - v = u_{2}$ , $u = \frac{1}{2} (u_{1} + u_{2})$ , $v = \frac{1}{2} (u_{1} - u_{2})$ , these equations take the form required.

Prob. 20.: In passing to circular functions, show that the only modiﬁcation to be made in the conversion formulas is in the algebraic sign of the right-hand member of the second formula.
Prob. 21.: Simplify $\frac{cosh 2 u + cosh 4 v}{sinh 2 u + sinh 4 v}$ , $\frac{cosh 2 u + cosh 4 v}{cosh 2 u - cosh 4 v}$ .
Prob. 22.: Prove ${sinh}^{2} x - {sinh}^{2} y = sinh (x + y) sinh (x - y)$ .
Prob. 23.: Simplify ${cosh}^{2} x {cosh}^{2} y \pm {sinh}^{2} x {sinh}^{2} y$ .
Prob. 24.: Simplify ${cosh}^{2} x {cos}^{2} y + {sinh}^{2} x {sin}^{2} y$ .

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Article 12Conversion Formulas.

Article 12
Conversion Formulas.