## 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 En 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 Series expansions in WolframAlpha.