Basic Trigonometric Identities

A trigonometric identity is an equation in trigonometric ratios that holds for every admissible value of the angle — not merely for some special value. For instance $\sec\theta = \dfrac{1}{\cos\theta}$ is an identity, true no matter what $\theta$ is (wherever $\cos\theta\ne0$). By contrast, $\sin\theta=\dfrac12$ is not an identity — it fails for almost every $\theta$ (e.g. at $\theta=60^\circ$) and is just an equation true for a few specific angles.

Identities are the working tools of trigonometry: they let a complicated trigonometric expression be rewritten as a simpler equivalent one, and they underlie almost every technique for solving trigonometric equations.

The three fundamental (Pythagorean) identities. Starting from the Pythagorean theorem applied to a right triangle and dividing through by the square of the hypotenuse, the adjacent side, and the opposite side in turn:

A few conventions and cautions:

• $\sin^2\theta$ is standard shorthand for $(\sin\theta)^2$ — never $\sin(\theta^2)$ — and likewise for every other trig ratio.
• $\sec^2\theta-\tan^2\theta=1$ has no meaning at $\theta=90^\circ$ (both terms are individually undefined there), yet it is still an identity because it holds at every $\theta$ where both sides are actually defined — an identity only needs to hold across its own domain, not literally everywhere.
• Likewise, $\dfrac{\sin\theta}{1+\cos\theta}$ is only meaningful where $1+\cos\theta \ne 0$. …