Multiple Angle Formulas formulas

Expressing trigonometric functions of angle $3\alpha$, $4\alpha$ and $5\alpha$ in terms of functions of an angle $\alpha$.

$\sin\left(3\alpha\right)=3\sin\alpha-4\sin^3\alpha$

$\sin\left(3\alpha\right)=3\cos^2\alpha \cdot \sin \alpha-\sin^3\alpha$

$\sin(4\alpha)=4\sin\alpha \cos\alpha -8\sin^3\alpha \cos\alpha$

$\sin \left(5\alpha\right) = 5\sin\alpha - 20\sin^3\alpha+16\sin^5\alpha$

$\cos3\alpha=4\cos^3\alpha-3\cos\alpha$

$\cos3\alpha=\cos^3\alpha-3\cos\alpha\sin^2\alpha$

$\cos4\alpha = 8\cos^4\alpha-8\cos^2\alpha+1$

$\cos5\alpha=16\cos^5\alpha-20\cos^3\alpha+5\cos\alpha$

$\tan3\alpha=\dfrac{3\tan \alpha-\tan^3\alpha}{1-3\tan^2\alpha}$

10.

$\cot3\alpha=\dfrac{\cot^3\alpha-3\cot\alpha}{3\cot^2\alpha-1}$