 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2023/paper-1/11/c/solution

For real

x

cosh x > 0

, so multiplication by the hyperbolic cosine is reversible. The identities

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

and

sech x = 1/ cosh x

give

e^{- x} cosh x = 2, \frac{1 + e ^{- 2 x}}{2} = 2.

Thus

e^{- 2 x} = 3

. The exponential function is injective, so the unique real solution is

x = - \frac{1}{2} ln 3.

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