Show that there is no polynomial $p ( x )$ for which $\cos ( \theta ) = p ( \sin \theta )$ for all angles $\theta$ in some nonempty interval. Hint: Note that $x$ and $| x |$ are different functions but their values are equal on an interval (as $x = | x |$ for all $x \geq 0$). You may want to show as a first step that this cannot happen for two polynomials, i.e., if polynomials $f$ and $g$ satisfy $f ( x ) = g ( x )$ for all $x$ in some interval, then $f$ and $g$ must be equal as polynomials, i.e., in each degree they must have the same coefficient.
Show that there is no polynomial $p ( x )$ for which $\cos ( \theta ) = p ( \sin \theta )$ for all angles $\theta$ in some nonempty interval.
Hint: Note that $x$ and $| x |$ are different functions but their values are equal on an interval (as $x = | x |$ for all $x \geq 0$). You may want to show as a first step that this cannot happen for two polynomials, i.e., if polynomials $f$ and $g$ satisfy $f ( x ) = g ( x )$ for all $x$ in some interval, then $f$ and $g$ must be equal as polynomials, i.e., in each degree they must have the same coefficient.