Let $a,b,c,d$ be real numbers. Show that the zeros of the functions $f(x)=a\cos x+b\sin x$ and $g(x)=c\cos x+d\sin x$ are distinct and alternate whenever $ad-bc\neq 0$.
Suppose $x_0\in \mathbb{R}$ such that $f(x_0)=g(x_0)=0$ we have $$\begin{bmatrix}a &b\\c&d\end{bmatrix}\begin{bmatrix}\cos x_0\\ \sin x_0\end{bmatrix}=\begin{bmatrix}0\\ 0\end{bmatrix}$$
As $\begin{bmatrix}a &b\\c&d\end{bmatrix}$ is with nonzero determinat $ad-bc$ we see that $\cos x_0=\sin x_0=0$ which is not possible as $\sin x$ and $\cos x$ can not be zero simultaneously.
Now, to show that zeros of $f(x)$ and $g(x)$ occur alternatively.
Suppose $x_0,x_1$ be zeros of $f(x)$ we then have by mean value theorem an element $z_0\in(x_0,x_1)$ such that $f'(z_0)=0$.. I thought of using something like this as derivative of $f$ and the function $g$ are closely related.
But it did not work.. Existence of $z_0\in(x_0,x_1)$ with $f'(z_0)=0$ is valid even then $f(x_0)\neq 0\neq f(x_1)$. So, I thought I am neglecting some crucial information by deducing this weak result..
Consider giving me some hints to solve this.
EDIT : I mean $f(x)$ and $g(x)$ have alternate zeros if between two zeros of $f(x)$ there is a zero of $g(x)$ and between two zeros of $g(x)$ there is a zero of $f(x)$.
Let $r=\sqrt{a^2+b^2}$, then $a=r\sin\varphi$ and $b=r\cos\varphi$ with some $\varphi$. Hence, $$ f(x) = a\cos x + b\sin x = r\sin\varphi \cdot \cos x + r\cos\varphi \cdot \sin x = r\cdot \sin(x+\varphi). $$ The roots of this function are $k\pi-\varphi$ (for $k\in\mathbb{Z}$); they form an arithmetic progression with difference $\pi$.
Similary, There are some real numbers $R>0$ and $\vartheta$ such that $c=R\sin\vartheta$ and $d=R\cos\vartheta$, so $$ g(x) = c\cos x + d\sin x = R\sin(x+\vartheta), $$ with roots $k\pi-\vartheta$ (for $k\in\mathbb{Z}$).
Moreover, $$ 0 \ne ad-bc = r\sin\varphi\cdot R\cos\vartheta - r\cos\varphi\cdot R\sin\vartheta = rR \cdot \sin(\varphi-\vartheta), $$ so the two arithmetic progression are distinct.
Both sets of roots are arithmetic progressions with the same difference and they are distinct, so they alternate.