$$\cos^2\alpha = \frac{1+\cos2\alpha}{2},$$

$$\sin^2\alpha = \frac{1-\cos2\alpha}{2},$$

$$tg\,^2\alpha = \frac{1-\cos2\alpha}{1+\cos2\alpha},$$

$$ctg\,^2\alpha = \frac{1+\cos2\alpha}{1-\cos2\alpha},$$

$$\cos^2\frac{\alpha}{2}=\frac{1+\cos \alpha}{2},$$

$$\sin^2\frac{\alpha}{2}=\frac{1-\cos \alpha}{2},$$

$$tg\,^2\frac{\alpha}{2}= \frac{1-\cos\alpha}{1+\cos\alpha},$$

$$ctg\,^2\frac{\alpha}{2}= \frac{1+\cos\alpha}{1-\cos\alpha},$$

$$\cos^3\alpha=\frac{3\cos\alpha+\cos 3\alpha}{4},$$

$$\sin^3\alpha=\frac{3\sin\alpha-\sin 3\alpha}{4},$$

$$tg\,^3\alpha= \frac{3\sin\alpha-\sin 3\alpha}{3\cos\alpha+\cos 3\alpha},$$

$$ctg\,^3\alpha= \frac{3\cos\alpha+\cos 3\alpha}{3\sin\alpha-\sin 3\alpha},$$

$$\cos^4\alpha = \frac{3+4\cos 2\alpha+\cos 4\alpha}{8},$$

$$\sin^4\alpha = \frac{3-4\cos 2\alpha+\cos 4\alpha}{8},$$

$$\cos^5\alpha = \frac{10\cos\alpha+5 \cos 3\alpha+\cos 5\alpha}{16},$$

$$\sin^5\alpha = \frac{10\sin\alpha-5\sin 3\alpha+\sin 5\alpha}{16}.$$

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