Reciprocal and Ratio Identities
Starting off with some really simple identities, these all follow straight from the definitions and require almost no proof.
Reciprocal Identities
\[\sin \theta = \frac{y}{r} = \frac{1}{r/y} = \frac{1}{\csc \theta}\] \[\cos \theta = \frac{x}{r} = \frac{1}{r/x} = \frac{1}{\sec \theta}\] \[\tan \theta = \frac{y}{x} = \frac{1}{x/y} = \frac{1}{\cot \theta}\]Ratio Identities
\[\tan \theta = \frac{y}{x} = \frac{y/r}{x/r} = \frac{\sin \theta}{\cos \theta} = \frac{1/\csc \theta}{1/\sec \theta} = \frac{\sec \theta}{\csc \theta}\] \[\cot \theta = \frac{x}{y} = \frac{x/r}{y/r} = \frac{\cos \theta}{\sin \theta} = \frac{1/\sec \theta}{1/\csc \theta} = \frac{\csc \theta}{\sec \theta}\]Now, once the values of $\sin(\theta)$ and $\cos(\theta)$ are defined, the values of the rest of the functions are forced.