Assume that the sine and cosine functions are continuous at the point 0.
(a) Find functions $a(x), b(h), c(x), d(h)$ such that $b(h)$ and $d(h)$ are continuous at $h = 0$ and $b(0) = d(0) = 0$, such that $|a(x)| ≤ 1$ and $|c(x)| ≤ 1$ for all $x$, and such that $cos(x + h) −cos(x) = a(x)b(h) + c(x)d(h)$ for all $x$ and $h$.
(b) Prove that the cosine function is continuous everywhere.
Shouldn't $a(x)$ and $c(x)$ be $1/x$ and then expand $cos(x + h)$ but then I would get $b$ and $d$ one in terms of other? how do I find $b$ and $d$?