 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/c/solution

Substitute the exponential definitions of hyperbolic cosine and hyperbolic sine. The cross terms cancel:

cosh x cosh y - sinh x sinh y = \frac{( e ^{x} + e ^{- x} ) ( e ^{y} + e ^{- y} ) - ( e ^{x} - e ^{- x} ) ( e ^{y} - e ^{- y} )}{4} = \frac{e ^{x - y} + e ^{- x + y}}{2} = cosh (x - y) .

This proves the identity for real arguments, and the same algebra proves it for complex arguments.

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