How to prove that cosine of a angle raised to a integer power can be expressed as below

Question

I wanted to know how to prove this relation, If possible. $$ \cos^k(\theta)\ =\ \sum_{j=-k}^k \left( a_j\,\cos(j\,\theta)\;+\;b_j\,\sin(j\,\theta) \right) $$ for some constants $a_j$, $b_j$ and integer $k$.
I got this far : $$\begin{align} \cos^k(\theta)\; & =\; \left(\frac{e^{i\theta}\; +\; e^{-i\theta}}{2} \right)^k \\ & =\; \frac{1}{2^k}\,\sum_{j=0}^k \left(\,\begin{pmatrix}k \\ j \end{pmatrix}\,e^{i(j)\theta}\;e^{-i(k-j)\theta\,} \right) \\ & =\; \frac{1}{2^k}\,\sum_{j=0}^k \left(\,\begin{pmatrix}k \\ j \end{pmatrix}\,e^{i(2j-k)\theta} \,\right) \\ & =\; \frac{e^{-ik\theta}}{2^k}\,\sum_{j=0}^k \left(\,\begin{pmatrix}k \\ j \end{pmatrix}\,e^{i(2j)\theta} \,\right) \\ & =\; \frac{e^{-ik\theta}}{2^k}\,\sum_{j=0}^k \left(\,\begin{pmatrix}k \\ j \end{pmatrix}\,\left(\,\cos(2j\theta)\,+\,i\sin(2j\theta) \,\right)\,\right) \end{align}$$ I could not figure out how to express this in the required form. Please include some steps so that I can understand with my high school maths.