## 9. Cotangents and tangents, cosecants and secants

### Cotangents

From the results obtained in the previous section we can easily deduce

Using the (3.9) to express the hyperbolic cotangent, we have

If we explicitly write some addends of the sum we have

Finally, using a more formal and compact notation:

The series expansion of the circular cotangent can be obtained from the (9.1),
(7.21) and (7.22)

In a more compact way:

### Tangents

To deduce the series expansion of the hyperbolic tangent we can use the equality

In fact

So, using twice the series expansion of the hyperbolic cotangent, we get

If we calculate explicitly the coefficients, we have

In a more compact way:

To have the series expansion of the circular tangent we use the equality

We get

If we calculate explicitly the coefficients, we have

In a more compact way:

### Cosecants

Since

the series expansion of the hyperbolic cosecant can be obtained using the series expansions of the hyperbolic tangent and cotangent.
From (9.1) and (9.3) we get

More explicitly

From (7.22) e (9.7) we get

and, in more compact way

### Secants

The hyperbolic secant can be expressed as a function of the exponential

This expression of the hyperbolic secant coincides with that of the function *s(x)* defined in (8.2) of the previous page,
then

where *E*_{n} is the *n*-th Euler's number.
Since, for odd indices these numbers are zero, we can more economically write

From the Euler's formulas, the circular secant can be expressed as

The even powers of the imaginary unit oscillate between 1 and -1, then