Tangent is a continuous strictly increasing bijection from $(-\pi \mathrm{/}2,\pi \mathrm{/}2)$ onto $\mathbb{R}$

Given: No hypotheses beyond those quantified in the statement.

Tangent is defined where cosine is nonzero and is differentiable on its natural domain with $(\tan x{)}^{\mathrm{\prime }}={\sec }^{2}x$; differentiability there implies continuity (Tangent, cotangent, secant, and cosecant on their exact natural domains, Derivatives and fundamental periods of tangent, cotangent, secant, and cosecant, A function differentiable at $c$ is continuous at $c$).

On the natural domain of tangent, ${\sec }^{2}x=1+{\tan }^{2}x$; moreover $\sec x=1\mathrm{/}\cos x\mathrm{\ne }0$, so ${\sec }^{2}x>0$ (Pythagorean and parity identities for all six trigonometric functions on their natural domains, Tangent, cotangent, secant, and cosecant on their exact natural domains, Squares of nonzero elements are positive).

The map $t\mapsto (\cos t,\sin t)$ maps $[0,2\pi )$ bijectively onto the unit circle ($t\mapsto (\cos t,\sin t)$ is a bijection from $[0,2\pi )$ onto the real unit circle).

Cosine decreases on $[0,\pi ]$, increases on $[\pi ,2\pi ]$, has zeros at $\pi \mathrm{/}2$ and $3\pi \mathrm{/}2$ in $[0,2\pi )$, and sine and cosine have period $2\pi$ (Signs, monotonicity intervals, and ranges of sine and cosine, The zero sets of sine and cosine and the least positive common period 2 pi).

Every nonnegative real has a unique nonnegative square root (Square roots exist: a unique $\sqrt{a}\ge 0$ with $(\sqrt{a}{)}^2=a$; the positives are $\{x^2:x\mathrm{\ne }0\}$).

A continuous function on an interval whose derivative is positive at every interior point is strictly increasing (On an interval $I$, for $f$ continuous on $I$ and differentiable at every interior point: $f^{\mathrm{\prime }}\ge 0$ throughout gives $f$ nondecreasing, $f^{\mathrm{\prime }}>0$ gives $f$ increasing, $f^{\mathrm{\prime }}\le 0$ and $f^{\mathrm{\prime }}<0$ give the two decreasing forms; conversely a nondecreasing $f$ has $f^{\mathrm{\prime }}\ge 0$ and a nonincreasing $f$ has $f^{\mathrm{\prime }}\le 0$ wherever it is differentiable, and no strict converse is claimed).