# Deriving Trigonometric Identities

We begin by recalling a very important identity from trigonometry,

$\cos^2{\theta} + \sin^2{\theta} = 1\,.$

Additionally, beginning from $$(1)$$ and dividing through by $$\cos^2{\theta}$$,

\begin{aligned} \frac{\cos^2{\theta}}{\cos^2{\theta}} + \frac{\sin^2{\theta}}{\cos^2{\theta}} &= \frac{1}{\cos^2{\theta}} \\ \\ 1 + \tan^2{\theta} &= \sec^2{\theta}\,. \\ \end{aligned}

Another set of identities worth knowing are,

\begin{aligned} \sin(\alpha \pm \beta) &= \sin \alpha \cos\beta \pm \sin\beta\cos\alpha \\ \cos(\alpha \mp \beta) &= \cos \alpha \cos\beta \pm \sin\beta\sin\alpha\,. \\ \end{aligned}

Recall that a circle of radius $$r$$ is described in cartesian coordinates as,

$x^2 + y^2 = r^2 \,.$

Extending from $$(4)$$, we can describe polar coordinates in terms of cartesian coordinates as follows.

\begin{aligned} x &= r \cos{\theta} \\ y &= r \sin{\theta} \\ r &= \sqrt{x^2 + y^2} \\ \tan\theta &= \frac{y}{x} \end{aligned}