6.23.19 Rewriting sin, cos, tan in terms of tan(x/2): halftan

The halftan command rewrites the trigonometric functions in terms of tan(x/2) using the identities:

• halftan takes one argument:
  expr, an expression containing trigonometric functions.
• halftan(expr) returns expr with any trigonometric functions replaced by the appropriate expression of tan(x/2).

Examples.

• Input:
  halftan(sin(2*x)/(1+cos(2*x)))
  Output:

  2tan ⎛ ⎜ ⎜ ⎝ 2 2  x ⎞ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎝ tan2 ⎛ ⎜ ⎜ ⎝ 2 2 x ⎞ ⎟ ⎟ ⎠ +1 ⎞ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1+ 1−tan2 ⎛ ⎜ ⎜ ⎝ 2 2 x ⎞ ⎟ ⎟ ⎠ tan2 ⎛ ⎜ ⎜ ⎝ 2 2 x ⎞ ⎟ ⎟ ⎠ +1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

  Output (after simplification with normal(ans())):

  tanx

• Input:
  halftan(sin(x)^2+cos(x)^2)
  Output:

  ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

  2 tan ⎛ ⎜ ⎜ ⎝ x 2 ⎞ ⎟ ⎟ ⎠ tan2 ⎛ ⎜ ⎜ ⎝ x 2 ⎞ ⎟ ⎟ ⎠ +1

  ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

  2

  +

  ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

  1−tan2 ⎛ ⎜ ⎜ ⎝ x 2 ⎞ ⎟ ⎟ ⎠ tan2 ⎛ ⎜ ⎜ ⎝ x 2 ⎞ ⎟ ⎟ ⎠ +1

  ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

  2

  Output (after simplification with normal(ans())):

  1