Trigonometrical identities

Here I show some useful trigonometric identities that are often used in Astronomy and other sciences

Pythagorean identities

$\sin^2 x + \cos^2 x = 1$

$\frac{1}{\tan^{2}x} = \sec^{2}x$

$\csc^{2}x = \frac{1}{\cot^{2}x}$

Ratio identities

$\tan x=\frac{\sin x}{\cos x}$

$\cot x=\frac{\cos x}{\sin x}$

Reciprocal identities

$\csc x=\frac{1}{\sin(x)}$

$\sec x=\frac{1}{\cos(x)}$

$\cot x=\frac{1}{\tan(x)}$

Opposite angles

$\sin (-x) = -\sin (x)$

$\cos (-x) = \cos (x)$

$\tan (-x) = -\tan (x)$

Works the same for the reciprocal identities.

Complementary angles

$\sin (90°-x) = \cos x$

$\cos (90°-x) = \sin x$

$\tan (90°-x) = -\cot x$

$\cot (90°-x) = -\tan x$

Supplementary angles

$\sin (180°-x) = \sin x$

$\cos (180°-x) = -\cos x$

$\tan (180°-x) = -\tan x$

$\cot (180°-x) = -\cot x$

Sum and Difference of Angles

$\sin(\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta$

$\cos(\alpha \pm \beta) = \sin \alpha \cos \beta \mp \cos \alpha \sin \beta$

$\tan(\alpha \pm \beta) = \frac{\tan \alpha \pm \tan \beta}{1 \mp \tan \alpha \tan \beta}$

Double angles

$\sin(2x) = 2\sin x \cos x$

$\cos(2x) = \cos^2 x - \sin^2 x$

$\tan(2x) = \frac{2\tan x}{1-\tan^2 x}$

Half angles

$\sin(\frac{x}{2}) = \pm \sqrt[]{\frac{1-\cos(x)}{2}}$

$\cos(\frac{x}{2}) = \pm \sqrt[]{\frac{1+\cos(x)}{2}}$

$\tan(\frac{x}{2}) = \frac{1 - \cos x}{\sin x}$