Fundamental (Pythagorean) Identities

The three Pythagorean (fundamental) identities connect the trigonometric ratios of the same angle and hold for every value of $\theta$ for which the ratios involved are defined. They follow directly from $x^2+y^2=r^2$ (Pythagoras' theorem applied to the point $P(x,y)$ on the terminal side, at distance $r$ from the origin).

Dividing $x^2+y^2=r^2$ throughout by $r^2$ gives

Dividing the same equation throughout by $x^2$ gives

Dividing throughout by $y^2$ gives

$1 + \cot^2\theta = \operatorname{cosec}^2\theta$ …