Change of variable for integrals involving trigonometric functions
In integral calculus, the tangent half-angle substitution is a change of variables used for evaluating integrals, which converts a rational function of trigonometric functions of formula_1 into an ordinary rational function of formula_2 by setting formula_3. This is the one-dimensional stereographic projection of the unit circle parametrized by angle measure onto the real line. The general transformation formula is:
formula_4
The tangent of half an angle is important in spherical trigonometry and was sometimes known in the 17th century as the half tangent or semi-tangent. Leonhard Euler used it to evaluate the integral formula_5 in his 1768 integral calculus textbook, and Adrien-Marie Legendre described the general method in 1817.
The substitution is described in most integral calculus textbooks since the late 19th century, usually without any special name. It is known in Russia as the universal trigonometric substitution, and also known by variant names such as "half-tangent substitution" or "half-angle substitution". It is sometimes misattributed as the Weierstrass substitution. Michael Spivak called it the "world's sneakiest substitution".
The substitution.
Introducing a new variable formula_6 sines and cosines can be expressed as rational functions of formula_7 and formula_8 can be expressed as the product of formula_9 and a rational function of formula_7 as follows:
formula_11
Similar expressions can be written for tan "x", cot "x", sec "x", and csc "x".
Derivation.
Using the double-angle formulas formula_12 and formula_13 and introducing denominators equal to one by the Pythagorean identity formula_14 results in
formula_15
Finally, since formula_16, differentiation rules imply
formula_17
and thus 
formula_18
Examples.
Antiderivative of cosecant.
formula_19
We can confirm the above result using a standard method of evaluating the cosecant integral by multiplying the numerator and denominator by formula_20 and performing the substitution formula_21 formula_22.
formula_23
These two answers are the same because formula_24
formula_25
The secant integral may be evaluated in a similar manner.
A definite integral.
formula_26
In the first line, one cannot simply substitute formula_27 for both limits of integration. The singularity (in this case, a vertical asymptote) of formula_28 at formula_29 must be taken into account. Alternatively, first evaluate the indefinite integral, then apply the boundary values.
formula_30
By symmetry,
formula_31
which is the same as the previous answer.
Third example: both sine and cosine.
formula_32
if formula_33
Geometry.
As "x" varies, the point (cos "x", sin "x") winds repeatedly around the unit circle centered at (0, 0). The point
formula_34
goes only once around the circle as "t" goes from −∞ to +∞, and never reaches the point (−1, 0), which is approached as a limit as "t" approaches ±∞. As "t" goes from −∞ to −1, the point determined by "t" goes through the part of the circle in the third quadrant, from (−1, 0) to (0, −1). As "t" goes from −1 to 0, the point follows the part of the circle in the fourth quadrant from (0, −1) to (1, 0). As "t" goes from 0 to 1, the point follows the part of the circle in the first quadrant from (1, 0) to (0, 1). Finally, as "t" goes from 1 to +∞, the point follows the part of the circle in the second quadrant from (0, 1) to (−1, 0).
Here is another geometric point of view. Draw the unit circle, and let "P" be the point (−1, 0). A line through "P" (except the vertical line) is determined by its slope. Furthermore, each of the lines (except the vertical line) intersects the unit circle in exactly two points, one of which is "P". This determines a function from points on the unit circle to slopes. The trigonometric functions determine a function from angles to points on the unit circle, and by combining these two functions we have a function from angles to slopes.
Hyperbolic functions.
As with other properties shared between the trigonometric functions and the hyperbolic functions, it is possible to use hyperbolic identities to construct a similar form of the substitution, formula_35:
formula_36
Similar expressions can be written for tanh "x", coth "x", sech "x", and csch "x". Geometrically, this change of variables is a one-dimensional stereographic projection of the hyperbolic line onto the real interval, analogous to the Poincaré disk model of the hyperbolic plane.
Alternatives.
There are other approaches to integrating trigonometric functions. For example, it can be helpful to rewrite trigonometric functions in terms of e"ix" and e−"ix" using Euler's formula.
Notes and references.
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