 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2003/paper-1/9/a/solution

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

Dividing these identities wherever

cos x \neq = 0

gives

tanh (i x) = i tan x

. The two undivided identities hold for every complex number

x

; the quotient identity excludes its poles.

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