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

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) =

e^x−e^−x

, cosh(x) =

e^x+e^−x

, tanh(x) =

e^x−e^−x

e^x+e^−x

e^2x−1

e^2x+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))

2tan

⎛
⎜
⎜
⎝

⎞
⎟
⎟
⎠

1−tan²

⎛
⎜
⎜
⎝

⎞
⎟
⎟
⎠

e^2 x−1

e^2 x+1

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