 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-2/13/a/solution

For

x > 0

, the exponential definitions of the hyperbolic functions give

cosh x - sinh x = e^{- x} > 0

and

sinh x > 0

. Taking reciprocals of these positive quantities proves

1/ cosh x < 1/ sinh x

. Also

cosh x > 1

for

x > 0

, so division by the positive

sinh x

gives

1/ sinh x < cosh x / sinh x

. Combining,

\frac{1}{cosh x} < \frac{1}{sinh x} < coth x (x > 0) .

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