Problem

(a) Use the definitions of hyperbolic functions in terms of exponentials to show that

$$\sinh(A+B)\equiv \sinh A\cosh B+\cosh A\sinh B$$

(b) Hence express $10\sinh x+8\cosh x$ in the form $R\sinh(x+\alpha)$ where $R>0$, giving $\alpha$ in the form $\ln p$ where $p$ is an integer.

(c) Hence solve the equation

$$10\sinh x+8\cosh x=18\sqrt7$$

giving your answer in the form $\ln(\sqrt7+q)$ where $q$ is a rational number to be determined.