Fundamental Trigonometric Identities

Before we build complex structures, we must understand our basic tools. Trigonometry is built on relationships. The Reciprocal Identities define the 'flip' of our primary functions: $\csc \theta = \frac{1}{\sin \theta}$, $\sec \theta = \frac{1}{\cos \theta}$, and $\cot \theta = \frac{1}{\tan \theta}$. Alongside these are the Quotient Identities, which reveal that $\tan \theta$ and $\cot \theta$ are actually ratios of sine and cosine: $\tan \theta = \frac{\sin \theta}{\cos \theta}$ and $\cot \theta = \frac{\cos \theta}{\sin \theta}$. Think of these as 'translation rules'—whenever you see a complex term like $\sec \theta$, you can translate it into the more familiar $\cos \theta$ to make the math easier to manage.

Simplifying trigonometric expressions is like solving a puzzle. The goal is to reduce a complex string of terms into a single term or a simple constant. A golden rule in simplification is to convert everything to sine and cosine. Because all other functions (tan, cot, sec, csc) are defined in terms of sine and cosine, this strategy often reveals hidden cancellations. Look for common denominators when adding fractions and always keep an eye out for the expression $\sin^2 \theta + \cos^2 \theta$, which you can immediately replace with the number 1.

Verifying an identity is different from solving an equation. You are given an equality and must prove that both sides are identical. The most important rule: work on only one side at a time. Usually, it is easier to start with the more complicated side and simplify it until it matches the simpler side. Common techniques include factoring, expanding brackets, or finding a common denominator for fractions. If you get stuck, try multiplying the numerator and denominator by a conjugate (e.g., if you have $1 - \sin \theta$, multiply by $1 + \sin \theta$) to create a Pythagorean identity.