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 $y$

$f(x+y)=f(x) f(y)-g(x) g(y)$ and

$g(x+y)=f(x) g(y)+g(x) f(y)$

If $$f^{\prime}(0)=0,$ prove that $(f(x))^{2}+(g(x))^{2}=1$ for all $x$$

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Let $$G(x)=\int_{0}^{x}\left[s \int_{0}^{s} f(t) d t\right] d s$$ where $f$ is continuous for all real $t$ . Find (a) $G(0),$ (b) $G^{\prime}(0)$(c) $G^{\prime \prime}(x),$ and $(\mathrm{d}) G^{\prime \prime}(0).$