Students usually learn the following basic table of tangent function values for special points of the circle:
For real values of argument , the values of are real.
In the points , the values of are algebraic. In several cases they can be integers , 0, or 1:
The values of can be expressed using only square roots if and is a product of a power of 2 and distinct Fermat primes {3, 5, 17, 257, …}.
The function is an analytical function of that is defined over the whole complex ‐plane and does not have branch cuts and branch points. It has an infinite set of singular points:
(a) are the simple poles with residues –1.
(b) is an essential singular point.
It is a periodic function with the real period :
The function is an odd function with mirror symmetry:
The first derivative of has simple representations using either the function or the function:
The derivative of has much more complicated representations than symbolic derivatives for and :
where is the Kronecker delta symbol: and .
The function satisfies the following firstorder nonlinear differential equation:
The function has a simple series expansion at the origin that converges for all finite values with :
where are the Bernoulli numbers.
The function has a wellknown integral representation through the following definite integral along the positive part of the real axis:
The function has the following simple continued fraction representations:
Indefinite integrals of expressions involving the tangent function can sometimes be expressed using elementary functions. However, special functions are frequently needed to express the results even when the integrands have a simple form (if they can be evaluated in closed form). Here are some examples:
Definite integrals that contain the tangent function are sometimes simple. For example, the famous Catalan constant can be defined as the value of the following integral:
This constant also appears in the following integral:
Some special functions can be used to evaluate more complicated definite integrals. For example, the generalized hypergeometric and polygamma functions are needed to express the following integral:
The following finite sums that contain the tangent function can be expressed using cotangent functions:
Other finite sums that contain the tangent function can be expressed using polynomial functions:
The evaluation limit of the first formula from the previous subsubsection for gives the following value for the corresponding infinite sum from the tangent:
Other infinite sums that contain the tangent can also be expressed using elementary functions:
The following finite product from the tangent has a very simple value:
The tangent of a sum can be represented by the rule: "the tangent of a sum is equal to the sum of tangents divided by one minus the product of tangents." A similar rule is valid for the tangent of the difference:
In the case of multiple arguments , , , …, the function can be represented as the ratio of the finite sums including powers of tangents:
The tangent of the half‐angle can be represented using two trigonometric functions by the following simple formulas:
The sine function in the last formula can be replaced by the cosine function. But it leads to a more complicated representation that is valid in some vertical strips:
To make this formula correct for all complex , a complicated prefactor is needed:
where contains the unit step, real part, imaginary part, the floor, and the round functions.
The sum of two tangent functions can be described by the rule: "the sum of tangents is equal to the sine of the sum multiplied by the secants." A similar rule is valid for the difference of two tangents:
The product of two tangent functions and the product of the tangent and cotangent have the following representations:
The most famous inequality for the tangent function is the following:
There are simple relations between the function and its inverse function :
The second formula is valid at least in the vertical strip . Outside of this strip a much more complicated relation (that contain the unit step, real part, and the floor functions) holds:
Tangent and cotangent functions are connected by a very simple formula that contains the linear function in the argument:
The tangent function can also be represented using other trigonometric functions by the following formulas:
The tangent function has representations using the hyperbolic functions:
The tangent function is used throughout mathematics, the exact sciences, and engineering.
