Formula for cos(k*x)

Question

I need to prove that: \begin{align} c_k =&\; \cos(k\!\cdot\!x)\\ c_k :=&\; c_{k-1} +d_{k-1}\\ d_k :=&\; 2d_0\!\cdot\!c_k +d_{k−1}\\ d_0 :=&\; −2\!\cdot\!\sin^2{(x/2)}\\ \end{align}

I've got an explicit formula for $d_k$ which should be:

\begin{align} d_k&=d_o+\sum_{i=1}^k{2\!\cdot\!d_o\!\cdot\!c_i} &&\implies& c_k &=c_{k-1}+ \sum_{i=1}^k{2\!\cdot\!d_o\!\cdot\!c_i} \end{align}

Now I want to do a proof by induction. Assuming that $c_p=\cos(p\!\cdot\!x)$ for every $p<k$. This would get me the following: $$c_k =\cos\left(\left(k-1\right)\!\cdot\!x\right)+\sum_{i=1}^k{2\!\cdot\!d_o\!\cdot\!\cos(p\!\cdot\!x)}$$ Using this formula I found:

\begin{align} \sum_{k=1}^n \cos(kx) & = \frac{\sin\left(\frac{nx}2\right)}{\sin \left(\frac{x}2\right)}\, \cos\left(\frac{(n+1)\,x}2\right) \end{align}

I tried to play around with trigonometric addition formulas but I am getting nowhere.

Answer 1

The last equation reads $$d_0=-2\sin^2\left(\frac x2\right)=\cos(x)-1.$$

Then eliminating $d$ from the second and the third,

$$d_k=c_{k+1}-c_k,\\ c_{k+1}-2c_k+c_{k-1}=2(\cos(x)-1)c_k.$$

If we assume that $c_k=\cos(kx)$, then after some simplification,

$$\cos(kx+k)-2\cos(kx)+\cos(kx-x)=2\cos(kx)\cos(x)-2\cos(kx)\\=2(\cos(x)-1)\cos(kx).$$

Hence $c_k=\cos(kx)$ is indeed a solution, such that $c_0=1$ (not specified in the OP).

We also have

$$d_k=\cos((k+1)x)-\cos(kx).$$

Now knowing these relations, formally proving by induction isn't a problem anymore.

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By the induction hypothesis, $$c_{k-1}=\cos((k-1)x),\\d_{k-1}=\cos(kx)-\cos((k-1)x).$$

Then,

$$\color{green}{c_k}=c_{k-1}+d_{k-1}=\cos((k-1)x)+\cos(kx)-\cos((k-1)x)\color{green}{=\cos(kx)}$$ and $$\color{green}{d_k}=2(\cos(x)-1)c_k+d_{k-1}\\ =2(\cos(x)-1)\cos(kx)+\cos(kx)-\cos((k-1)x)\\ =2\cos(x)\cos(kx)-2\cos(kx)+\cos(kx)-\cos(kx)\cos(x)-\sin(kx)\sin(x)\\ =\cos(kx)\cos(x)-\sin(kx)\sin(x)-\cos(kx)\\ \color{green}{=\cos((k+1)x)-\cos(kx)}.$$

And the base case is

$$\color{green}{c_0}=\cos(0\cdot x)=\color{green}{1},\\ \color{green}{d_0}=\cos(1\cdot x)-\cos(0\cdot x)=\color{green}{\cos(x)-1}.$$