$$1+\cos\alpha=2\cos^2\frac{\alpha}{2}$$
$$1-\cos\alpha=2\sin^2\frac{\alpha}{2}$$
$$1+\sin\alpha=2\cos^2\left (\frac{\pi}{2}-\frac{\alpha}{2} \right )$$
$$1-\sin\alpha=2\sin^2\left (\frac{\pi}{2}-\frac{\alpha}{2} \right )$$
$$1\pm\text{tg}\,\alpha\;\text{tg}\,\beta=\frac{\cos(\alpha\mp\beta)}{\cos\alpha\cos\beta}$$
$$\text{ctg}\,\alpha\;\text{ctg}\,\beta\pm1=\frac{\cos(\alpha\mp\beta)}{\sin\alpha\sin\beta}$$
$$1\pm\text{tg}\,\alpha=\frac{\sqrt{2}\sin(45^{\circ}\pm\alpha)}{\cos\alpha}$$
$$1-\text{tg}^2\,\alpha=\frac{\cos2\alpha}{\cos^2\alpha}$$
$$1-\text{ctg}^2\,\alpha=-\frac{\cos2\alpha}{\sin^2\alpha}$$
$$\text{tg}^2\,\alpha-\text{tg}^2\,\beta=\frac{\sin(\alpha+\beta)\sin(\alpha-\beta)}{\cos^2\alpha\cos^2\beta}$$
$$\text{ctg}^2\,\alpha-\text{ctg}^2\,\beta=\frac{\sin(\alpha+\beta)\sin(\beta-\alpha)}{\sin^2\alpha\sin^2\beta}$$
$$\text{tg}^2\,\alpha-\sin^2\alpha=\text{tg}^2\,\alpha\cdot\sin^2\alpha$$
$$\text{ctg}^2\,\alpha-\cos^2\alpha=\text{ctg}^2\,\alpha\cdot\cos^2\alpha$$