Quotient Identity
Quotient identities express tangent as sine divided by cosine, and cotangent as cosine divided by sine.
Formula
\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}; \quad \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}
Definition
Quotient identities say that tangent equals sine divided by cosine, and cotangent equals cosine divided by sine: $\tan(\theta) = \sin(\theta)/\cos(\theta)$ and $\cot(\theta) = \cos(\theta)/\sin(\theta)$. They follow from the SOH-CAH-TOA definitions, $\tan = \text{opposite}/\text{adjacent} = (\text{opposite}/\text{hypotenuse})/(\text{adjacent}/\text{hypotenuse})$, and geometrically, for a point $(x, y) = (\cos(\theta), \sin(\theta))$ on the unit circle, the slope of the line from the origin to $(x, y)$ is $y/x = \sin/\cos = \tan$, linking tangent to slope and to the tangent line in differential geometry. As a ratio of entire functions, $\tan(x)$ is meromorphic on $\mathbb{C}$ with simple poles at $x = \pi/2 + n\pi$.
Example
If $\sin(\theta) = 0.6$ and $\cos(\theta) = 0.8$, then $\tan(\theta) = 0.6/0.8 = 0.75$, matching opposite/adjacent directly. Simplifying $(\sin^2(x) + \cos^2(x))/\cos^2(x) = 1/\cos^2(x) = \sec^2(x)$, or equivalently $\tan^2(x) + 1 = \sec^2(x)$, recovers the Pythagorean identity. The addition formula $\tan(a + b) = (\tan(a) + \tan(b))/(1 - \tan(a)\tan(b))$ is derived from the quotient identity combined with the sine and cosine addition formulas, dividing numerator and denominator by $\cos(a)\cos(b)$.
Key Insight
These identities show that you only really need sine and cosine: all other trig functions can be built from those two by division or reciprocals, and this is why $\tan(90^\circ)$ is undefined, since $\sin(90^\circ) = 1$ but $\cos(90^\circ) = 0$. The quotient identity is an instance of the general principle that ratios of holomorphic functions are meromorphic, so tan's zeros and poles encode the complete analytic structure of both sin and cos simultaneously.