Let S and C be two functions. Assume that the domain for...

Let $S$ and $C$ be two functions. Assume that the domain for both $S$ and $C$ is the set of all real numbers and that $S$ and C satisfy the following two identities. $$ \begin{aligned} S(x-y) &=S(x) C(y)-C(x) S(y) \\ C(x-y) &=C(x) C(y)+S(x) S(y) \end{aligned} $$ Also, suppose that the function $S$ is not identically zero. That is, $S(x) \neq 0 \quad$ for at least one real number $x$ (a) Show that $S(0)=0 .$ Hint: In identity (1), let $x=y$ (b) Show that $C(0)=1 .$ Hint: In identity (1), let $y=0$ (c) Explain (in complete sentences) why it was necessary to use condition (3) in the work for part (b). (d) Prove the identity $[C(x)]^{2}+[S(x)]^{2}=1$ Hint: In identity (2), let $y=x$ (e) Show that $C$ is an even function and $S$ is an odd function. That is, prove the identities $C(-x)=C(x)$ and $S(-x)=-S(x)$ Hint: Write $-x$ as $0-x$

Let $S$ and $C$ be two functions. Assume that the domain for both $S$ and $C$ is the set of all real numbers and that $S$ and C satisfy the following two identities.
$$
\begin{aligned}
S(x-y) &=S(x) C(y)-C(x) S(y) \\
C(x-y) &=C(x) C(y)+S(x) S(y)
\end{aligned}
$$
Also, suppose that the function $S$ is not identically zero. That is,
$S(x) \neq 0 \quad$ for at least one real number $x$
(a) Show that $S(0)=0 .$ Hint: In identity (1), let $x=y$
(b) Show that $C(0)=1 .$ Hint: In identity (1), let $y=0$
(c) Explain (in complete sentences) why it was necessary to use condition (3) in the work for part (b).
(d) Prove the identity $[C(x)]^{2}+[S(x)]^{2}=1$ Hint: In identity (2), let $y=x$
(e) Show that $C$ is an even function and $S$ is an odd function. That is, prove the identities $C(-x)=C(x)$ and $S(-x)=-S(x)$
Hint: Write $-x$ as $0-x$

Key Concepts

Functional Equations

This concept involves equations in which the unknowns are functions and the equations describe relationships between their values at various points. In the context of this problem, the identities provided for S and C are functional equations, and solving them involves determining the functions’ behaviors by strategically plugging in specific values (like setting x=y or y=0) to simplify and deduce properties.

Trigonometric Identities

The given identities mirror the classical addition formulas for sine and cosine. Recognizing this resemblance is central to the problem as it motivates proving that S and C behave like sine and cosine functions, respectively. This connection helps in anticipating properties such as S(0)=0, C(0)=1, and the Pythagorean identity, which are analogous to standard trigonometric identities.

Identity Element (Normalization)

In many functional equations, especially those modeling trigonometric functions, the value at zero plays the role of an identity element. Here, determining that S(0)=0 and C(0)=1 is critical as it provides a normalization or base case that anchors the behavior of the functions and ensures consistency with the known properties of sine and cosine. This step is essential for deriving further properties and identities.

Even and Odd Functions

The concepts of even and odd functions relate to symmetry properties of functions, where an even function is symmetric about the y-axis (f(-x)=f(x)) and an odd function is symmetric about the origin (f(-x)=-f(x)). In this problem, proving that C is even and S is odd is a direct consequence of applying the provided identities to negative arguments, reflecting the inherent symmetries observed in cosine and sine functions.

Nontriviality Condition

The condition that S is not identically zero (i.e., S(x) ? 0 for at least one x) is crucial because without it, the functions could degenerate into trivial solutions that do not resemble familiar trigonometric functions. This nontriviality ensures that meaningful properties like C(0)=1 are derived from the identities, by ruling out the possibility that all observed equalities result from the zero function, thereby validating the normalization step in the argument.

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