<s>[INST] How compute $\cos(5\theta)$ and $\sin(5\theta)$? I would like to compute $\cos(5\theta)$ and $\sin(5\theta)$. I can use the formula $\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)$ and $\sin(a+b)=\sin(a)\cos(b)+cos(a)\sin(b)$ but it's a little bit to long. Is there an other way to compute it ?
 [/INST] \begin{align}
c_2 &=c^2-s^2, \hspace{5mm} s_2=2cs \\
c_4 &= c_2^2-s_2^2 = (c^2-s^2)^2-4c^2s^2 = c^4-6c^2s^2+s^4,\\
s_4 &= 2c_2s_2 = 4cs(c^2-s^2) = 4c^3s-4cs^3. \\
c_5 &= c_4c-s_4s = (c^4-6c^2s^2+s^4)c-(4c^3s-4cs^3)s = c^5-10c^3s^2+5cs^4,\\ s_5 &= c_4s+s_4c = (c^4-6c^2s^2+s^4)s+(4c^3s-4cs^3)c = 5c^4s-10c^2s^3+s^5.
\end{align}
</s>