# Double and Multiple Angle Formulas

Angles (arguments of functions): $$\alpha$$
Trigonometric functions: $$\sin \alpha,$$ $$\cos \alpha,$$ $$\tan \alpha,$$ $$\cot \alpha$$
1. Sine of a double angle
$$\sin 2\alpha = 2\sin \alpha \cos \alpha$$
2. Cosine of a double angle
$$\cos 2\alpha = {\cos ^2}\alpha – {\sin ^2}\alpha =$$ $$1 – 2\,{\sin ^2}\alpha =$$ $$2\,{\cos ^2}\alpha – 1$$
3. Tangent of a double angle
$$\tan 2\alpha = {\large\frac{{2\tan \alpha }}{{1 – {{\tan }^2}\alpha}}\normalsize} =$$ $${\large\frac{2}{{\cot \alpha – \tan \alpha }}\normalsize}$$
4. Cotangent of a double angle
$$\cot 2\alpha = {\large\frac{{{{\cot }^2}\alpha – 1}}{{2\cot \alpha}}\normalsize} =$$ $${\large\frac{{\cot \alpha – \tan \alpha }}{2}\normalsize}$$
5. Sine of a triple angle
$${\sin 3\alpha = 3\sin\alpha – 4\,{\sin^3} \alpha }=$$ $$3\,{\cos ^2}\alpha \sin \alpha – {\sin ^3}\alpha$$
6. $$\sin 4\alpha = 4\sin \alpha \cos \alpha \,-$$ $$8\,{\sin ^3}\alpha \cos \alpha$$
7. $$\sin 5\alpha = 5\sin\alpha \,-$$ $$20\,{\sin ^3}\alpha \,+$$ $$16\,{\sin ^5}\alpha$$
8. Cosine of a triple angle
$$\cos 3\alpha =$$ $$4\,{\cos^3} \alpha – 3\cos \alpha =$$ $${\cos ^3}\alpha – 3\,{\sin ^2}\alpha \cos \alpha$$
9. $$\cos 4\alpha = 8\,{\cos ^4}\alpha \,-$$ $$8\,{\cos ^2}\alpha + 1$$
10. $$\cos 5\alpha = 16\,{\cos ^5}\alpha \,-$$ $$20\,{\cos ^3}\alpha \,+$$ $$5\cos \alpha$$
11. Tangent of a triple angle
$$\tan 3\alpha =$$ $$\large\frac{{3\tan \alpha – {{\tan }^3}\alpha }}{{1 – 3\,{{\tan }^2}\alpha }}\normalsize$$
12. $$\tan 4\alpha =$$ $$\large\frac{{4\tan \alpha – 4\,{{\tan }^3}\alpha }}{{1 – 6\,{{\tan }^2}\alpha + \,{{\tan }^4}\alpha }}\normalsize$$
13. $$\tan 5\alpha =$$ $$\large\frac{{\,{{\tan }^5}\alpha – 10\,{{\tan }^3}\alpha + 5\tan \alpha }}{{1 – 10\,{{\tan }^2}\alpha + 5\,{{\tan }^4}\alpha }}\normalsize$$
14. Cotangent of a triple angle
$$\cot 3\alpha =$$ $$\large\frac{{\,{\cot^3}\alpha – 3\cot \alpha }}{{3\,{{\cot }^2}\alpha – 1}}\normalsize$$
15. $$\cot 4\alpha =$$ $$\large\frac{{1 – 6\,{{\tan }^2}\alpha + {{\tan }^4}\alpha }}{{4\tan \alpha – 4\,{{\tan }^3}\alpha}}\normalsize$$
16. $$\cot 5\alpha =$$ $$\large\frac{{1 – 10\,{{\tan }^2}\alpha + 5\,{{\tan }^4}\alpha }}{{\,{{\tan }^5}\alpha – 10\,{{\tan }^3}\alpha + 5\tan \alpha }}\normalsize$$