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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## Recommended Questions

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$ e = \dfrac{271,801}{99,990} $

If $(-4,-5)$ is a point on a graph that is symmetric with respect to the $y$ -axis, then $(4,-5)$ is also a point on the graph.

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$\frac{sin\ 60^\circ}{sin\ 30^\circ} = sin\ 2^\circ$

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True or False? In Exercises $63-68$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

\begin{equation}

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\end{equation}

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137. Proof Use the Product Rule twice to prove that if $f, g,$ and

$h$ are differentiable functions of $x,$ then

$$\frac{d}{d x}[f(x) g(x) h(x)]=f^{\prime}(x) g(x) h(x)+f(x) g^{\prime}(x) h(x)+f(x) g(x) h^{\prime}(x)$$

True or False? In Exercises $83-86$ , determine whether the statement is true or false. If it is false, explain why or give anexample that shows it is false.

$$\arcsin ^{2} x+\arccos ^{2} x=1$$

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$$\int \sin ^{2} 2 x \cos 2 x d x=\frac{1}{3} \sin ^{3} 2 x+C$$

$sin\ 60^\circ$ $csc\ 60^\circ$ $= 1$

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The relative maxima of the function $f$ are $f(1)=4$ and $f(3)=10 .$ Therefore, $f$ has at least one minimum for some $x$ in the interval $(1,3) .$

True or False? In Exercises $83-86$ , determine whether the

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example that shows it is false.

Every rational function has a slant asymptote.

True or False? In Exercises $93-98$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

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$$f(a)=f(b), \text { then } a=b$$

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