Increasing the phase by π/2 in a trigonometric expression

The shift_phase command increases the phase of a trigonometric expression by π/2.

shift_phase takes expr, a trigonometric expression.
shift_phase(expr) returns expr with the phase increased by π/2 (after automatic simplification).

shift_phase(x+sin(x))

x−cos

⎛
⎜
⎜
⎝

π +2 x

⎞
⎟
⎟
⎠

shift_phase(x+cos(x))

x+sin

⎛
⎜
⎜
⎝

π +2 x

⎞
⎟
⎟
⎠

shift_phase(x+tan(x))

x−

tan

⎛
⎜
⎜
⎝

π +2 x

⎞
⎟
⎟
⎠

Quoting the argument will prevent the automatic simplification.

shift_phase('sin(x+pi/2)')

−cos

⎛
⎜
⎜
⎝

π +2 x+2

⎞
⎟
⎟
⎠

With an unquoted sine, you get:

shift_phase(sin(x+pi/2))

sin

⎛
⎜
⎜
⎝

π +2 x

⎞
⎟
⎟
⎠

since sin(x+pi/2) is evaluated (in this case simplified) before shift_phase is called, and shift_phase(cos(x)) returns sin((pi+2*x)/2).