\textbf{Triple angle formulas:}

,,\qquad \sin(3\alpha) = 3\sin(\alpha) - 4\sin^3(\alpha)
,,\qquad \cos(3\alpha) = 4\cos(\alpha) - 3\cos(\alpha)

>What about quadruple and quintuple angle formulas?
Coefficients for n-angle formulas are the coefficients of \href{https://en.wikipedia.org/wiki/Chebyshev_polynomials}{Chebyshev polynomials}.
Specifically Chebyshev polynomials of the first kind for sine, and Chebyshev polynomials of the second kind for cosine.

Triple angle formula for tangnet: 

,,\qquad \tan(3\alpha) = \frac{3\tan(\alpha) - \tan^3{\alpha}}{1 - 3\tan^2{\alpha}}