10.2.19 Rewriting sin, cos, tan in terms of half tangent
The
halftan
command rewrites the trigonometric functions in terms of tan(
x
/2) using the identities:
sin(
x
) =
2tan
⎛
⎜
⎜
⎝
x
2
⎞
⎟
⎟
⎠
tan
2
⎛
⎜
⎜
⎝
x
2
⎞
⎟
⎟
⎠
+1
, cos(
x
) =
1−tan
2
⎛
⎜
⎜
⎝
x
2
⎞
⎟
⎟
⎠
tan
2
⎛
⎜
⎜
⎝
x
2
⎞
⎟
⎟
⎠
+1
, tan(
x
) =
2tan
⎛
⎜
⎜
⎝
x
2
⎞
⎟
⎟
⎠
1−tan
2
⎛
⎜
⎜
⎝
x
2
⎞
⎟
⎟
⎠
.
halftan
takes
expr
, an expression containing trigonometric functions.
halftan(
expr
)
returns
expr
with any trigonometric functions replaced by the appropriate expression of
tan(
x
/2)
.
Examples
halftan
(
sin
(2*
x
)/(1+
cos
(2*
x
)))
2
tan
⎛
⎜
⎜
⎝
2
2
x
⎞
⎟
⎟
⎠
⎛
⎜
⎜
⎝
tan
2
⎛
⎜
⎜
⎝
2
2
x
⎞
⎟
⎟
⎠
+1
⎞
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎜
⎜
⎝
1+
1−
tan
2
⎛
⎜
⎜
⎝
2
2
x
⎞
⎟
⎟
⎠
tan
2
⎛
⎜
⎜
⎝
2
2
x
⎞
⎟
⎟
⎠
+1
⎞
⎟
⎟
⎟
⎟
⎟
⎠
normal
(
ans
())
tan
x
halftan
(
sin
(
x
)^2+
cos
(
x
)^2)
⎛
⎜
⎜
⎜
⎜
⎜
⎝
2
tan
⎛
⎜
⎜
⎝
x
2
⎞
⎟
⎟
⎠
tan
2
⎛
⎜
⎜
⎝
x
2
⎞
⎟
⎟
⎠
+1
⎞
⎟
⎟
⎟
⎟
⎟
⎠
2
+
⎛
⎜
⎜
⎜
⎜
⎜
⎝
1−
tan
2
⎛
⎜
⎜
⎝
x
2
⎞
⎟
⎟
⎠
tan
2
⎛
⎜
⎜
⎝
x
2
⎞
⎟
⎟
⎠
+1
⎞
⎟
⎟
⎟
⎟
⎟
⎠
2
normal
(
ans
())
1
