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

## Discussion

## Video Transcript

No transcript available

## Recommended Questions

True or False? In Exercises $73-78$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

$\int f(x) g(x) d x=\left(\int f(x) d x\right)\left(\int g(x) d x\right)$

True or False? In Exercises $73-78$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $F(x)$ and $G(x)$ are antiderivatives of $f(x),$ then

$$F(x)=G(x)+C$$

True or False? In Exercises $85-90$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $f^{\prime}(x)=g^{\prime}(x),$ then $f(x)=g(x)$

True or False? In Exercises $85-90$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $f(x)=-g(x)+b,$ then $f^{\prime}(x)=-g^{\prime}(x)$

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

If

$$\int _ { a } ^ { b } [ f ( x ) - g ( x ) ] d x = A$$

then

$$\int _ { a } ^ { b } [ g ( x ) - f ( x ) ] d x = - A$$

True or False? In Exercises $73-76$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $$\lim _{x \rightarrow c} f(x)=L,$ then $f(c)=L$$

Give the chemical names of each of the following familiar compounds:$(\mathbf{a})$$\mathrm{NaCl}($ table salt $)$ $(\mathbf{b})$$\mathrm{NaHCO}_{3}$ (baking soda),$(\mathbf{c}) \mathrm{NaOCl}$ (in many branches)$(\mathbf{d})$ $\mathrm{NaOH}($ caustic soda $)$ $(\mathbf{e})$ $\left(\mathrm{NH}_{4}\right)_{2} \mathrm{CO}_{3}($ smelling salts $)$$(\mathbf{f})$$\mathrm{CaSO}_{4}$ (plaster of Paris).

The floor of a railroad flatcar is loaded with loose crates having a coefficient of static friction of 0.25 with the floor. If the train

is initially moving at a speed of 48 $\mathrm{km} / \mathrm{h}$ , in how short a distance

can the train be stopped at constant acceleration without causing

the crates to slide over the floor?