Trigonometric identities

1) Even/Odd function identities

$$\sin(-\alpha) = -\sin\alpha \tag{1.1}$$
$$\cos(-\alpha) = \cos\alpha \tag{1.2}$$
$$\tan(-\alpha) = -\tan\alpha \tag{1.3}$$
$$\cot(-\alpha) = -\cot\alpha \tag{1.4}$$

2) Sum/Addition identities

$$\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta \tag{2.1}$$
$$\sin(\alpha - \beta) = \sin\alpha\cos\beta - \cos\alpha\sin\beta \tag{2.2}$$
$$\cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta \tag{2.3}$$
$$\cos(\alpha - \beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta \tag{2.4}$$

$$\tan(\alpha + \beta) = \frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}\tag{2.5}$$

$$\cot(\alpha + \beta) = \frac{\cot\alpha\cot\beta-1}{\cot\alpha+\cot\beta}\tag{2.6}$$

3) Sum to product identities

$$ \sin\alpha + \sin\beta = 2 \sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2} \tag{3.1}$$
$$ \sin\alpha - \sin\beta = 2 \sin\frac{\alpha-\beta}{2}\cos\frac{\alpha+\beta}{2} \tag{3.2}$$

$$ \cos\alpha + \cos\beta = 2 \cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2} \tag{3.3}$$

$$ \cos\alpha - \cos\beta = -2 \sin\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2} \tag{3.4}$$

$$ \tan\alpha \pm \tan\beta = \frac{\sin{(\alpha\pm\beta)}}{\cos\alpha\cos\beta} \tag{3.5}$$

$$ \cot\alpha \pm \cot\beta = \frac{\sin{(\beta\pm\alpha)}}{\sin\alpha\sin\beta} \tag{3.6}$$

4) Product to sum identities

$$\sin\alpha\sin\beta = \frac{1}{2}[\cos(\alpha-\beta) - \cos(\alpha+\beta)] \tag{4.1}$$

$$\cos\alpha\cos\beta = \frac{1}{2}[\cos(\alpha-\beta) + \cos(\alpha+\beta)] \tag{4.2}$$

$$\sin\alpha\cos\beta = \frac{1}{2}[\sin(\alpha+\beta) + \sin(\alpha-\beta)] \tag{4.3}$$

5) Double-angle and triple-angle identities

$$\sin 2 \alpha  = 2 \sin\alpha \cos\alpha \tag{5.1}$$

$$\cos 2 \alpha  = \cos^2\alpha - \sin^2\alpha \tag{5.2}$$

$$\tan 2 \alpha  = \frac{2\tan\alpha}{1-\tan^2\alpha} = \frac{2}{\cot\alpha - \tan\alpha} \tag{5.3}$$

$$\cot 2 \alpha  = \frac{\cot^2\alpha-1}{2\cot\alpha} = \frac{\cot\alpha - \tan\alpha}{2} \tag{5.4}$$

$$\sin 3 \alpha  = 3 \sin\alpha - 4\sin^3 \alpha \tag{5.5}$$

$$\cos 3 \alpha  = 4\cos^3 \alpha - 3 \cos\alpha \tag{5.6}$$

$$\tan 3 \alpha  = \frac{3\tan \alpha - \tan^3 \alpha}{1 - 3 \tan^2 \alpha} \tag{5.7}$$

$$\cot 3 \alpha  = \frac{\cot^3 \alpha - 3 \cot \alpha}{3\cot^2 \alpha - 1} \tag{5.8}$$

6) Decreasing the power of sine and cosine

$$2 \sin^2 \alpha = 1 - \cos 2\alpha \tag{6.1}$$

$$4 \sin^3 \alpha = 3 \sin \alpha - \sin 3\alpha \tag{6.2}$$

$$8 \sin^4 \alpha = \cos 4\alpha - 4 \cos 2\alpha + 3 \tag{6.3}$$

$$2 \cos^2 \alpha = 1 + \cos 2\alpha \tag{6.4}$$

$$4 \cos^3 \alpha = 3 \cos \alpha + \cos 3\alpha \tag{6.5}$$

$$8 \cos^4 \alpha = \cos 4\alpha + 4 \cos 2\alpha + 3 \tag{6.6}$$