Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2003 / Paper 1 / 9

The exponential function definitions of the hyperbolic functions and Euler's formula giveDividing these identities wherever $\cos x\ne 0$ gives $\boxed{\tanh (ix)=i\tan x}$. The two undivided identities hold for every complex number $x$; the quotient identity excludes its poles.

Adding and subtracting the exponential function power series cancels the appropriate odd or even terms:Both series converge absolutely for every complex number $x$, as follows from the ratio test.

Two applications of integration by parts give $I=\cosh x\sin x-\int \sinh x\sin xdx$ and $\int \sinh x\sin xdx=-\sinh x\cos x+I$. Thus an antiderivative is $[\cosh x\sin x+\sinh x\cos x]/2$. Evaluating its endpoints gives