 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2009/paper-1/14/d/solution

The exponential definitions give

2 sinh x cosh x = \frac{( e ^{x} - e ^{- x} ) ( e ^{x} + e ^{- x} )}{2} = \frac{e ^{2 x} - e ^{- 2 x}}{2} = sinh (2 x) .

Thus the double-angle identity follows directly, without any restriction beyond the domain of the exponential.

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