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Question 1

1. Let $\cosh x=(e^{x}+e^{-x})/2$ and $\sinh x=(e^{x}-e^{-x})/2$ .

(i) Show that $\cosh^{2}x-\sinh^{2}x=1$ ; deduce that

\coth^{2}x={\hbox{cosech}}^{2}\,x+1.

(ii) Show that the inverse function of $\coth x$ is given by

\coth^{-1}x={{1}\over{2}}\log{{x+1}\over{x-1}}\qquad(x>1).

(iii) Calculate the derivatives of $\coth x$ and the inverse function $\coth^{-1}x$ .

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