# Rewriting trigonometric functions in terms of tan(x/2) and hyperbolic functions in terms of exp(x): halftan_hyp2exp

### 5.23.20 Rewriting trigonometric functions in terms of tan(x/2) and hyperbolic functions in terms of exp(x): halftan_hyp2exp

The halftan_hyp2exp command rewrites the trigonometric function in terms of tan(x/2) (like halftan, see Section 5.23.19) 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 one argument:
  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.

- Input:
  halftan_hyp2exp(tan(x)+tanh(x))
  Output:

  2tan

  ⎛
  ⎜
  ⎜
  ⎝

  x
  2

  ⎞
  ⎟
  ⎟
  ⎠

  1−tan2

  ⎛
  ⎜
  ⎜
  ⎝

  x
  2

  ⎞
  ⎟
  ⎟
  ⎠

  +

  e2 x−1
  e2 x+1

- Input:
  halftan_hyp2exp(sin(x)^2+cos(x)^2-sinh(x)^2+cosh(x)^2)
  Output (after simplification with normal(ans())):

  2