Quotient Identity
TrigonometryQuotient 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.
Example
If $\sin(\theta) = 0.6$ and $\cos(\theta) = 0.8$, then $\tan(\theta) = 0.6/0.8 = 0.75$ (which matches using opposite/adjacent directly).
Key Insight
These identities show that you only really need to know sine and cosine. All other trig functions can be built from those two using division or reciprocals.
Definition
The quotient identities are $\tan(\theta) = \sin(\theta)/\cos(\theta)$ and $\cot(\theta) = \cos(\theta)/\sin(\theta)$. They follow from the SOH-CAH-TOA definitions: $\tan = \text{opposite}/\text{adjacent} = (\text{opposite}/\text{hypotenuse})/(\text{adjacent}/\text{hypotenuse}) = \sin/\cos$.
Example
Simplify: $(\sin^2(x) + \cos^2(x))/\cos^2(x) = 1/\cos^2(x) = \sec^2(x)$. Alternatively: $\sin^2/\cos^2 + 1 = \tan^2(x) + 1 = \sec^2(x)$, recovering the Pythagorean identity.
Key Insight
The quotient identity $\tan = \sin/\cos$ is used constantly in calculus to convert between trig functions. It also explains why $\tan(90^\circ)$ is undefined: $\sin(90^\circ) = 1$ but $\cos(90^\circ) = 0$, so the quotient has a zero denominator.
Definition
$\tan(x) = \sin(x)/\cos(x)$ is a meromorphic function on $\mathbb{C}$ with simple poles at $x = \pi/2 + n\pi$. The quotient identity is a consequence of the unit-circle definition: for point $(x, y) = (\cos(\theta), \sin(\theta))$, the slope of the line from the origin to $(x, y)$ is $y/x = \sin/\cos = \tan$. This geometric interpretation links tangent to slope, making it fundamental in differential geometry (the tangent line).
Example
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: $\sin(a+b)/\cos(a+b) = (\sin(a)\cos(b)+\cos(a)\sin(b))/(\cos(a)\cos(b)-\sin(a)\sin(b))$, dividing numerator and denominator by $\cos(a)\cos(b)$.
Key Insight
The quotient identity is an instance of the general principle that ratios of holomorphic functions are meromorphic. $\tan = \sin/\cos$ is entire sin divided by entire cos, giving a meromorphic function whose zeros and poles encode the complete analytic structure of both functions simultaneously.