Fundamental Identities

Fundamental Identities

A trigonometric identity is an equation that holds for every admissible value of θ — not just at a few special angles. All three fundamental identities trace back to the single geometric fact that a point $(\cos\theta,\sin\theta)$ always lies on the unit circle, i.e. $x^2+y^2=1$ with $x=\cos\theta,\ y=\sin\theta$:

This single relation already lets us solve for one function in terms of the other:

(the sign chosen according to which quadrant θ lies in).

Dividing identity (1) throughout by $\cos^2\theta$ (valid wherever $\cos\theta\ne0$) gives $\dfrac{\sin^2\theta}{\cos^2\theta}+1 = \dfrac1{\cos^2\theta}$, i.e.

Dividing identity (1) instead by $\sin^2\theta$ (valid wherever $\sin\theta\ne0$) gives $1+\dfrac{\cos^2\theta}{\sin^2\theta} = \dfrac1{\sin^2\theta}$, i.e.

$\textbf{3)}\quad 1+\cot^2\theta = \text{cosec}^2\theta$ …