### 10.2.20 Rewriting trigonometric/hyperbolic functions in terms of half tangent/exponentials

The halftan_hyp2exp command rewrites the trigonometric function in terms of tan(x/2) (like halftan, see Section 10.2.19) and rewrites the hyperbolic functions in terms of their definitions using exponentials, namely:

sinh(x) =

ex−e−x
2

,  cosh(x) =

ex+e−x
2

,  tanh(x) =

ex−e−x
ex+e−x

=

e2x−1
e2x+1

.

- halftan_hyp2exp takes expr, a trigonometric and hyperbolic expression.
- halftan_hyp2exp(expr) returns expr with any trigonometric functions replaced by the appropriate expression in tan(x/2) and any hyperbolic functions replaced by the appropriate exponentials.

#### Examples

halftan_hyp2exp(tan(x)+tanh(x))

2 tan

⎛
⎜
⎜
⎝

x
2

⎞
⎟
⎟
⎠

1−tan2

⎛
⎜
⎜
⎝

x
2

⎞
⎟
⎟
⎠

+

e2x−1
e2x+1

normal(halftan_hyp2exp(sin(x)^2+cos(x)^2-sinh(x)^2+cosh(x)^2))

2 |  |  |  |  |  |  |  |  |  |