1991 Putnam B2

Suppose $f$ and $g$ are non-constant, differentiable, real-valued

functions defined on $(-\infty, \infty)$. Furthermore, suppose that for

each pair of real numbers $x$ and $y$,

$$

\begin{aligned}

f(x+y) &= f(x)f(y) - g(x)g(y), \\

g(x+y) &= f(x)g(y) + g(x)f(y).

\end{aligned}

$$

If $f'(0) = 0$, prove that $(f(x))^2 + (g(x))^2 = 1$ for all $x$.

假设 $f$ 和 $g$ 是非常量、可微、实值

在 $(-\infty, \infty)$ 上定义的函数。此外，假设对于

每对实数 $x$ 和 $y$，

$$

\开始{对齐}

f(x+y) &= f(x)f(y) - g(x)g(y), \\

g(x+y) &= f(x)g(y) + g(x)f(y)。

\结束{对齐}

$$

如果 $f'(0) = 0$，则证明对于所有 $x$，$(f(x))^2 + (g(x))^2 = 1$。