Tangent half-angle formula

The tangent half-angle formula, also known as the Weierstrass substitution, is a trigonometric substitution technique in calculus that employs t = \tan(\theta/2) to express \sin \theta, \cos \theta, and d\theta as rational functions of t, thereby converting integrals involving rational combinations of sine and cosine into integrals of rational functions that can be evaluated using partial fractions.^[1] The core identities of the substitution are derived from double-angle formulas and are given by

\sin \theta = \frac{2t}{1 + t^2}, \quad \cos \theta = \frac{1 - t^2}{1 + t^2}, \quad d\theta = \frac{2 \, dt}{1 + t^2},

where t = \tan(\theta/2).^[1] These expressions transform an integral of the form \int f(\sin \theta, \cos \theta) \, d\theta into \int f\left( \frac{2t}{1+t^2}, \frac{1-t^2}{1+t^2} \right) \frac{2 \, dt}{1+t^2}, simplifying the computation for many indefinite and definite integrals in trigonometric form.^[1] The method is particularly effective for integrals that do not yield easily to other substitutions like those based on secant or tangent of the full angle.^[2] Named after the German mathematician Karl Weierstrass (1815–1897), who first employed it systematically in the 19th century for integrating rational trigonometric functions, the substitution has roots in earlier geometric parametrizations but was formalized in modern calculus contexts during that period.^[3] Geometrically, it provides a rational parametrization of the unit circle, where the point (\cos \theta, \sin \theta) traces the circle via the line through (-1, 0) with slope t, offering a stereographic-like projection that avoids irrationalities in coordinates and connects to classical problems like generating Pythagorean triples.^[4] This dual algebraic and geometric utility has made it a staple in advanced integration techniques and symbolic computation.^[1]

Definition and Formulae

Core Substitution Formula

The tangent half-angle substitution, commonly referred to as the Weierstrass substitution, defines the parameter t = \tan(\theta/2), where \theta represents the angle and t serves as the substitution variable.^[1] This approach yields a rational parametrization of the unit circle, allowing trigonometric functions to be expressed as rational functions of t.^[1] The core formulae derived from this substitution are:

\sin \theta = \frac{2t}{1 + t^2}

\cos \theta = \frac{1 - t^2}{1 + t^2}

\tan \theta = \frac{2t}{1 - t^2}

^[1] The corresponding differential form is d\theta = \frac{2 \, dt}{1 + t^2}.^[5] As t varies over all real numbers t \in \mathbb{R}, it maps to angles \theta \in (-\pi, \pi), excluding \theta = \pi where the substitution becomes undefined.^[5]

Derived Trigonometric Identities

The tangent half-angle substitution, with t = \tan(\theta/2), yields expressions for the reciprocal trigonometric functions by taking reciprocals of the fundamental sine and cosine formulas. Specifically,

\sec \theta = \frac{1 + t^2}{1 - t^2},

obtained as the reciprocal of \cos \theta = (1 - t^2)/(1 + t^2). Similarly,

\csc \theta = \frac{1 + t^2}{2t},

derived from the reciprocal of \sin \theta = 2t/(1 + t^2). And

\cot \theta = \frac{1 - t^2}{2t},

following from the ratio \cos \theta / \sin \theta. These identities facilitate the rationalization of trigonometric expressions involving reciprocals.^[6] The substitution directly encodes the double-angle formula for tangent, since \tan \theta = 2t / (1 - t^2), where t = \tan(\theta/2). This relation extends iteratively to multiple angles: for instance, applying the double-angle formula repeatedly expresses \tan(n \phi) as a rational function of t = \tan \phi, where \phi = \theta/2 and n is a positive integer. Such iterations are useful for deriving closed-form expressions for tangent of multiple half-angles. Power-reduction formulae also emerge from the substitution. For example, substituting into the double-angle identity for cosine gives

\cos(2\theta) = \frac{t^4 - 6t^2 + 1}{(1 + t^2)^2},

where t = \tan(\theta/2), providing a rational expression that reduces powers of cosine in terms of the half-angle parameter. This approach generalizes to higher even multiples, yielding polynomial relations over quadratic denominators. In the complex plane, the substitution links to exponential forms via stereographic projection. Combining the expressions for sine and cosine yields

e^{i\theta} = \cos \theta + i \sin \theta = \frac{1 + i t}{1 - i t},

a Möbius transformation that maps the parameter t on the real line to points on the unit circle in the complex plane. This representation underscores the substitution's role in unifying trigonometric and complex analytic perspectives. The identities exhibit limitations due to singularities in the substitution. Notably, at t = \pm 1, corresponding to \theta = \pm \pi/2 + 2k\pi for integer k, the denominator $1 - t^2 = 0, rendering \sec \theta and \tan \theta undefined, while \csc \theta and \cot \theta remain finite but the overall mapping breaks at these poles.

Proofs of the Formulae

Algebraic Derivation

The tangent half-angle formulas begin with the standard expressions for \tan(\theta/2) in terms of the full-angle trigonometric functions, derived from the double-angle identities. Specifically, using the double-angle formula for sine, \sin \theta = 2 \sin(\theta/2) \cos(\theta/2), and for cosine, \cos \theta = 1 - 2 \sin^2(\theta/2), it follows that

\tan\left(\frac{\theta}{2}\right) = \frac{\sin \theta}{1 + \cos \theta}.

To derive this, substitute the double-angle expressions into the right-hand side:

\frac{\sin \theta}{1 + \cos \theta} = \frac{2 \sin(\theta/2) \cos(\theta/2)}{1 + [1 - 2 \sin^2(\theta/2)]} = \frac{2 \sin(\theta/2) \cos(\theta/2)}{2 \cos^2(\theta/2)} = \frac{\sin(\theta/2)}{\cos(\theta/2)} = \tan\left(\frac{\theta}{2}\right).

Similarly, using \cos \theta = 2 \cos^2(\theta/2) - 1 and \sin \theta = 2 \sin(\theta/2) \cos(\theta/2), the equivalent form

\tan\left(\frac{\theta}{2}\right) = \frac{1 - \cos \theta}{\sin \theta}

is obtained by

\frac{1 - \cos \theta}{\sin \theta} = \frac{2 \sin^2(\theta/2)}{2 \sin(\theta/2) \cos(\theta/2)} = \frac{\sin(\theta/2)}{\cos(\theta/2)} = \tan\left(\frac{\theta}{2}\right).

Let t = \tan(\theta/2). To express \sin \theta and \cos \theta in terms of t, start with the double-angle formula for sine:

\sin \theta = 2 \sin(\theta/2) \cos(\theta/2) = 2 \tan(\theta/2) \cos^2(\theta/2) = 2 t \cos^2(\theta/2).

Since \sec^2(\theta/2) = 1 + \tan^2(\theta/2), it follows that \cos^2(\theta/2) = 1 / (1 + t^2). Thus,

\sin \theta = \frac{2 t}{1 + t^2}.

For cosine, apply the double-angle formula \cos \theta = \cos^2(\theta/2) - \sin^2(\theta/2):

\cos \theta = \cos^2(\theta/2) (1 - \tan^2(\theta/2)) = \frac{1}{1 + t^2} (1 - t^2) = \frac{1 - t^2}{1 + t^2}.

Alternatively, using \cos \theta = 1 - 2 \sin^2(\theta/2),

\cos \theta = 1 - 2 t^2 \cdot \frac{1}{1 + t^2} = \frac{1 + t^2 - 2 t^2}{1 + t^2} = \frac{1 - t^2}{1 + t^2},

confirming the result. The expression for \tan \theta follows directly from the double-angle formula for tangent:

\tan \theta = \frac{2 \tan(\theta/2)}{1 - \tan^2(\theta/2)} = \frac{2 t}{1 - t^2}.

This verifies consistency with the earlier expressions, as \tan \theta = \sin \theta / \cos \theta = [2 t / (1 + t^2)] / [(1 - t^2)/(1 + t^2)] = 2 t / (1 - t^2). Finally, to confirm the differential form, differentiate \theta = 2 \arctan t with respect to t:

\frac{d\theta}{dt} = 2 \cdot \frac{1}{1 + t^2},

so d\theta = 2 dt / (1 + t^2). This completes the algebraic derivation of the tangent half-angle substitution formulas.

Geometric Interpretation

The tangent half-angle formula offers a geometric parametrization of points on the unit circle x^2 + y^2 = 1, where the parameter t = \tan(\theta/2) corresponds to the angle \theta from the positive x-axis. The point (\cos \theta, \sin \theta) can be visualized by constructing a line passing through the fixed point (-1, 0) on the circle with slope t. This line intersects the unit circle at a second point, which is precisely (\cos \theta, \sin \theta). To derive the coordinates geometrically, substitute the line equation y = t(x + 1) into the circle equation, yielding the quadratic x^2(1 + t^2) + 2t^2 x + t^2 - 1 = 0. One root is x = -1, and the other is x = (1 - t^2)/(1 + t^2), with y = 2t/(1 + t^2). This construction highlights how the slope t directly encodes the half-angle, providing a rational parametrization that maps the real line (via t) to the circle excluding the point (-1, 0). This parametrization corresponds to the stereographic projection from the point (-1, 0) onto the y-axis. A line drawn from (-1, 0) through a point (\cos \theta, \sin \theta) on the circle intersects the y-axis at (0, t), where t = \tan(\theta/2). The formulas x = (1 - t^2)/(1 + t^2) and y = 2t/(1 + t^2) emerge naturally from this projection, illustrating how the tangent half-angle formula bridges angular measures on the circle to linear coordinates on the line. This projection preserves angles and provides a conformal map, underscoring the formula's role in transforming circular geometry into affine space. The geometric significance extends to the arc length element on the unit circle. Differentiating \theta = 2 \arctan t gives d\theta = 2 \, dt / (1 + t^2), where d\theta represents the infinitesimal arc length ds (since the radius is 1). This relation shows how increments in the parameter t correspond to arc lengths along the circle, with the factor $1/(1 + t^2) reflecting the distortion inherent in the stereographic map. Historically, such geometric constructions for parametrizing and dividing the circle trace back to ancient Greek astronomers, who used chord-based methods to compute arc divisions and table values; Hipparchus (c. 190–120 BCE) employed half-angle relations in chord tables for astronomical calculations, while Ptolemy (c. 100–170 CE) refined these in the Almagest using geometric intersections to derive arc lengths and angles.^[7]

Applications in Calculus

Weierstrass Substitution for Integration

The Weierstrass substitution provides a systematic method to evaluate indefinite integrals of the form \int R(\sin \theta, \cos \theta) \, d\theta, where R is a rational function. By introducing the variable t = \tan(\theta/2), the expressions \sin \theta = \frac{2t}{1 + t^2}, \cos \theta = \frac{1 - t^2}{1 + t^2}, and d\theta = \frac{2 \, dt}{1 + t^2} are substituted, converting the original trigonometric integral into an integral of a rational function in t. This rational function can then be decomposed using partial fractions and integrated using standard algebraic techniques, yielding an antiderivative expressible in terms of elementary functions.^[1] A classic example is the integral \int \frac{d\theta}{a + b \cos \theta} for a > |b|. Applying the substitution gives

\int \frac{d\theta}{a + b \cos \theta} = \int \frac{\frac{2 \, dt}{1 + t^2}}{a + b \cdot \frac{1 - t^2}{1 + t^2}} = \int \frac{2 \, dt}{(a + b) + (a - b) t^2}.

This simplifies to a standard arctangent form:

\frac{2}{\sqrt{a^2 - b^2}} \arctan\left( \frac{\sqrt{a^2 - b^2} \, t}{a + b} \right) + C = \frac{2}{\sqrt{a^2 - b^2}} \arctan\left( \frac{(a - b) \tan(\theta/2)}{\sqrt{a^2 - b^2}} \right) + C,

assuming a > b > 0. The partial fraction decomposition of the rational integrand in t directly leads to this result after integration. For definite integrals over a full period such as [0, 2\pi), the substitution t = \tan(\theta/2) maps the interval to the real line from -\infty to \infty, though a singularity occurs at \theta = \pi where t \to \pm \infty. The integral becomes \int_{-\infty}^{\infty} \frac{2 \, dt}{(a + b) + (a - b) t^2}, which evaluates to \frac{2\pi}{\sqrt{a^2 - b^2}} for a > b > 0, accounting for the principal value and the behavior at infinity (equivalent to a residue contribution of zero at t = \infty in the complex plane for the rational function). This approach avoids direct complex analysis while leveraging the symmetry of the substitution. The primary advantage of the Weierstrass substitution lies in its universality: it reduces a broad class of trigonometric integrals to algebraic ones amenable to partial fractions, often simplifying computations that would otherwise require specialized identities or numerical methods. It is especially effective for integrals like \int \sin^m \theta \cos^n \theta \, d\theta where m and n are odd integers or rational numbers, as the powers transform into ratios of polynomials in t of degree at most m + n + 1, which can be integrated routinely after decomposition. For instance, when one exponent is odd, the substitution complements reduction formulas by providing an explicit rational form.^[1]

Connections to Hyperbolic Functions

The hyperbolic analog of the tangent half-angle substitution employs u = \tanh(\phi/2), which rationalizes expressions involving hyperbolic functions into algebraic forms suitable for integration or other manipulations. With this substitution, the double-angle identities yield

\sinh \phi = \frac{2u}{1 - u^2}, \quad \cosh \phi = \frac{1 + u^2}{1 - u^2},

and the differential becomes

d\phi = \frac{2\, du}{1 - u^2}.

These relations parallel the trigonometric expressions \sin \theta = 2t/(1 + t^2) and \cos \theta = (1 - t^2)/(1 + t^2) where t = \tan(\theta/2), but with sign changes reflecting the hyperbolic identity \cosh^2 \phi - \sinh^2 \phi = 1.^[8] The connection between the trigonometric and hyperbolic substitutions stems from the analytic continuation of functions via imaginary arguments, governed by identities such as \tan(i\phi) = i \tanh \phi. Setting \theta = i\phi maps \tan(\theta/2) \leftrightarrow i \tanh(\phi/2), establishing a direct isomorphism that transfers identities and techniques between the domains. A prominent example is the pair of Pythagorean-like relations: $1 + \tan^2(\theta/2) = \sec^2(\theta/2) in the trigonometric case and $1 - \tanh^2(\phi/2) = \sech^2(\phi/2) in the hyperbolic case, highlighting the structural similarity with the sign flip arising from the imaginary linkage.^[9] This substitution finds applications in evaluating definite classes of hyperbolic integrals, such as \int \frac{d\phi}{a + b \cosh \phi} (assuming a > b > 0), where substituting u = \tanh(\phi/2) transforms the integrand into a rational function \frac{2\, du}{(a + b)(1 - u^2) + 2b u^2}, amenable to partial fraction decomposition. The resulting antiderivative often involves inverse tangents or logarithms, providing closed forms for otherwise intractable expressions. Beyond integrals, the substitution aids in solving nonlinear differential equations featuring hyperbolic terms, such as those modeling wave propagation or pendulum motion in relativistic contexts, by reducing them to algebraic equations. It also connects to special functions, including elliptic integrals, where analogous rationalizations bridge hyperbolic and trigonometric representations for numerical evaluation or asymptotic analysis.^[10]

The Gudermannian Function

The Gudermannian function, denoted \operatorname{gd}(x), provides a direct link between trigonometric and hyperbolic functions, serving as a bridge that maps hyperbolic arguments to circular angles without invoking complex numbers. It is defined as

\operatorname{gd}(x) = \int_0^x \operatorname{sech} t \, \mathrm{d}t, \quad -\infty < x < \infty.

Equivalent expressions include \operatorname{gd}(x) = 2 \arctan(\tanh(x/2)), \operatorname{gd}(x) = \arcsin(\tanh x), and \operatorname{gd}(x) = \arctan(\sinh x).^[11] This function was originally introduced by Johann Heinrich Lambert in the 1760s and named after the German mathematician Christoph Gudermann, who in his 1830 paper on potential or cyclically hyperbolic functions explored connections between elliptic, hyperbolic, and circular functions to unify their theories.^[12] ^[13] The inverse Gudermannian function, \operatorname{gd}^{-1}(\theta), reverses this mapping and is given by \operatorname{gd}^{-1}(\theta) = 2 \artanh(\tan(\theta/2)) for -\pi/2 < \theta < \pi/2, explicitly relating the hyperbolic argument to the tangent half-angle substitution t = \tan(\theta/2). Key properties include \operatorname{gd}(0) = 0 and \operatorname{gd}'(x) = \operatorname{sech} x; in the complex plane, it exhibits analytic continuation with branch points but inherits quasi-periodic behavior from the underlying hyperbolic functions, such as shifts by $2\pi i. Applications of the Gudermannian function arise in contexts requiring the interplay of circular and hyperbolic geometries, such as simplifying distance formulas and angle relations in hyperbolic geometry via its inverse. In special relativity, it facilitates the parameterization of Lorentz boosts by relating the unbounded rapidity (a hyperbolic angle) to bounded trigonometric angles, aiding derivations of velocity addition and spacetime transformations. ^[14] ^[15] ^[16] ^[17] ^[18]

Rational Points and Pythagorean Triples

The tangent half-angle substitution provides a parametrization of points on the unit circle x^2 + y^2 = 1 using a rational parameter t = \tan(\theta/2), where \theta is the angle corresponding to the point (\cos \theta, \sin \theta). Specifically, for a rational t = m/n in lowest terms with integers m, n > 0, the coordinates are given by

\cos \theta = \frac{n^2 - m^2}{n^2 + m^2}, \quad \sin \theta = \frac{2mn}{n^2 + m^2},

yielding a rational point on the unit circle since both components are rational.^[19]^[20] This parametrization directly connects to Pythagorean triples, as the rational point (\cos \theta, \sin \theta) scales to an integer triple (a, b, c) satisfying a^2 + b^2 = c^2 by multiplying through by the common denominator d = n^2 + m^2, giving a = |n^2 - m^2|, b = 2mn, c = n^2 + m^2. For primitive triples, where \gcd(a, b, c) = 1, select m > n > 0 coprime integers of opposite parity (one even, one odd), ensuring no common factor divides all three. For example, with m=2, n=1, the triple is (3, 4, 5).^[19]^[20]^[21] All primitive Pythagorean triples arise uniquely from this construction up to swapping a and b, as the formulas generate every such triple when m and n satisfy the conditions. Non-primitive triples are obtained by scaling primitives by a positive integer k > 1, yielding (ka, kb, kc), and the parametrization is complete in that every rational point on the unit circle (except (-1, 0), corresponding to t = \infty) is generated by some rational t.^[19]^[20] Geometrically, rational values of t represent lines of rational slope passing through the point (-1, 0) on the unit circle, with the second intersection point being rational; these slopes correspond to rational tangent half-angles, systematically covering all rational points except (-1, 0).^[19]^[20]