Past exam of the year 1 of the computer science course of the University of Oxford / 2018 / Functional programming, design and analysis of algorithms / 2 / h / Solution

The derivative relations and initial values give $S^{\prime }=C$, $S(0)=0$, $C^{\prime }=-S$ and $C(0)=1$. Integrating from zero yields $S={\int }_0^{x}C$ and $C=1-{\int }_0^{x}S$. Thus use mutually recursive lazy lists:

sinx, cosx :: Fractional a => [a]
sinx = integral cosx
cosx = minus one (integral sinx)

Both heads are available immediately, and later coefficients depend on earlier coefficients of the other stream. The resulting formal power series are $S=x-x^3/3!+x^5/5!-\cdots$ and $C=1-x^2/2!+x^4/4!-\cdots$. Their zero coefficients and alternating signs follow directly from the coupled recurrence relations, providing a coefficientwise proof without assumptions about analytic convergence.